
Glass. 
Book 



COPYRIGHT DEPOSIT 




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ELEMENTS 



OF 



PLANE SURVEYING 

t 

(INCLUDING LEVELING) 



BY 



SAMUEL MARX BARTON, Ph.D. 

professor of mathematics, university of the south; sometime professor 
of mathematics and civil engineering, virginia polytechnic 
institute; author of "an elementary treatise 
on the theory of equations " 



REVISED 



D. C. HEATH & CO., PUBLISHERS 
BOSTON NEW YORK CHICAGO 



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Copyright, 1904 and 1913, 
By D. C. Heath & Co. 

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PREFACE 

The primary object of this work is to give a brief treatise on 
Plane Surveying, adapted for a short course in colleges, or as a 
preparatory course in technological institutions. It has been the 
aim of the author to make clear points that, in his fifteen years' 
experience, both in the class-room and the field, he has found 
troublesome to beginners. 

In dealing with fundamental principles and operations he has 
endeavored to answer "simple questions that confound." In carry- 
ing this idea out, he has put into the book a few things that he has 
looked for in vain in existing text-books. The work is brief, and 
yet is not an outline for a lecture course. On the contrary, it is a 
book that might be studied privately, or be used by a teacher of but 
little practical experience. While the work is intended primarily 
as a text, in its preparation the author has kept in view the possi- 
bility of its falling into the hands of county surveyors who may not 
have had the advantages of a collegiate course. 

While the work is elementary, it is believed that it will be found 
to be scientific, and that the user of the book will not learn anything 
that he will have to unlearn in more advanced study of surveying 
or engineering. 

Especial attention is called to the following points : 

1. Careful description of instruments and their adjustments. 

2. The explicit statement of the methods of making a re-survey, 
in accordance with the different data to be had. 

3. The discussion of the declination of the needle, and the 
excellent isogonic chart at the beginning of the book. 

4. The simple methods of obtaining a true meridian line. 

5. Suggestive forms for field notes and for purposes of reduction. 

6. Many illustrative examples. 

7. The clear and complete set of tables. 

As the advantages of stadia work are becoming more and more 
recognized, the fundamental proofs of the theory are given together 



IV PREFACE 

with forms for the field notes. In the closing chapter plane leveling 
is treated in a manner and scope sufficient, it is thought, for the 
needs of the general surveyor. 

In the preparation of the book the standard American works 
have been consulted. 

The author takes this opportunity of thanking Professor Webster 
Wells for his courtesy in allowing him to reproduce his excellent 
six-place logarithmic tables and his table of the natural trigono- 
metric functions, which are given here as Tables XVII, XVIII, and 
XIX, respectively. He acknowledges his indebtedness to the U. S. 
Coast and Geodetic Survey for courtesies extended through Super- 
intendent O. H. Tittman. 

He desires also to acknowledge his indebtedness to the following 
instrument-makers for use of their cuts or diagrams : Messrs. W. & 
L. E. Gurley, Troy, N.Y. ; George N. Saegmuller, Washington, D.C. ; 
Young & Sons, Philadelphia; Brown & Sharpe Mfg. Co., Provi- 
dence, R. I.; and Eugene Dietzgen Co., Chicago. 

SAMUEL M. BARTON. 



REVISED EDITION 

In making this revision, the Tables giving the position of Polaris, 
etc., have been brought up to date ; some minor changes have been 
made in the chapter on the Declination of the Needle ; and a new 
isogonic map, compiled for the year 1910, has been put in the place 
of one for an earlier date. To make the matter on adjustments 
somewhat clearer and more in accordance with practice, two new 
pages have been added. 

The author takes this opportunity to thank Professor B. L. Coul- 
son for helpful suggestions. 

S. M. B. 
Sewanee, Tennessee, 
May, 1913. 



ANALYTICAL TABLE OF CONTENTS 

[The numbers refer to articles] 

INTRODUCTION 

Definitions: Surveying 1-4 

The Earth 5-12 

The Celestial Sphere 13-31 

CHAPTER I 

INSTRUMENTS— THEIR USE AND ADJUSTMENTS 

Gunter's Chain 32 

Engineer's Chain 33 

Tapes 34 

Standards of Length 35 

Comparison of the Chain and Tape 36 

Pins and Range Poles 37, 38 

To measure a Line with a Chain 39 

Errors, Compensating and Cumulative 40 

Correction for Length 41 

Correction for Area 42 

The Surveyor's Compass * 43-45 

Verniers 46-50 

Adjustments of the Compass 51-55 

The Use of the Compass 56-59 

The Solar Compass 60 

The Transit : 

Description of 61-63 

Adjustments of 64-71 

To measure a Horizontal Angle with 72 

To measure a Vertical Angle 73 

To prolong a Straight Line 74 

Solar Attachment : 

Description of 75, 76 

Theory of 77 

Adjustments of 78 

Saegmuller Solar Attachment : 

Description of 79 

Adjustments of 80 

Directions for using .81 

V 



vi ANALYTICAL TABLE OF CONTENTS 

Observation for Time 82 

To obtain Latitude 83 

Object of Solar Instrument 84, 85 

To find the Declination of the Sun ......... 86 

Correction for Refraction . . . . . 87 

Declination Settings 88 

To run Lines with the Solar Transit ......... 89 

Merits and Defects of Solar Instruments 90 

Levels 91 

The Y-Level 92 

Adjustments of Y-Level . 93-95 

Leveling-rods . 96-98 

The Plane-Table ... 99, 100 

The Sextant 101 

Drawing Instruments . 102-105 

Exercises. 

CHAPTER II 

CHAIN SURVEYING 

To range out a Line . . 106-108 

To erect a Perpendicular at Any Point of a Line 109 

To let fall a Perpendicular from a Given Point to a Given Line . . . 110, 111 
To run a Line Parallel to a Given Line . . . . . . . . .112 

Obstacles to Alinement and Measurement ....... 114-116 

Areas of Simple Figures ........... 117 

Heights .118 

Exercises. 

CHAPTER. Ill 

COMPASS SURVEYING 

I. Field Operations — Original Surveys and Re-surveys 

Angle between Two Courses 120, 121 

Field Work and Field Notes 122-124 

Local Attraction 125, 126 

General Example of a Survey of a Farm ........ 127 

Report of a Survey 128 

Surveying Party 129 

Re-surveys . 130-134 

II. The Declination op the Needle 

Definition of Declination and Variation 135 

Agonic Line, East and West Declination . 136 

Historic Note on the Magnetic Needle 137 

Irregular and Daily Variation 138 

Secular Variation 139 

To determine the Declination 140 



ANALYTICAL TABLE OF CONTENTS 



vn 



III. The Determination or a True Meridian Line 

The Problem stated, Azimuths of Polaris, Time of Elongations .... 141 

Reduction of Time to Other Dates 142 

Note on Civil and Astronomical Time. 
Illustrative Example. 

First Method of determining a True Meridian Line 143, 144 

Second Method 145 

Third Method 146 

Determination of Latitude 147 



CHAPTER IV 

COMPUTATION OF AREAS 

Definitions 

Formulae for Latitude and Departure 

The Traverse Table 

Error of Closure 

Balancing the Survey 

Double Meridian Distances 

Formula for Areas . 

Example 



148, 149 
. 150 
. 151 
. 152 

153-155 
. 156 
. 157 
. 158 



SUPPLYING OMISSIONS 

The Problems stated " . . . . . . .159 

The Bearing and Length of One Course omitted 160 

The Bearing of One Course and the Length of Another omitted .... 161 

The Bearings of Two Courses omitted 162 

The Lengths of Two Courses omitted . 163 

Examples. 

Coordinate Method of computing Areas 164 

Plotting . . . 165-167 

Copying ............... 168 

Graphic Method of computing Areas 169 

Examples. 

CHAPTER V 

TRANSIT SURVEYING 

I. Field Work and Computation of Areas 

Azimuths 171, 172 

To survey a Farm with a Transit 173, 174 

Azimuths changed into Bearings . . . . . . . . . .175 

Computation of Areas, Reduction Forms ....... 176, 177 

Interior Angles 178 

Deflection Angles, Field Notes . . . 179, 180 

Examples. 



II. Laying out and Dividing Land 



Laying out Land in Special Shapes 
Dividing Land . 



181-186 
187-190 



viii ANALYTICAL TABLE OF CONTENTS 

III. Stadia Surveying 

Fundamental Theory 191-196 

Formulae for Inclined Sights 197 

Stadia Tables . 198 

Accuracy of Stadia Measurements 199-202 

Form for Field Notes ............ 203 

IV. Public Lands of the United States 
Methods, Instruments, Regulations . . . . . . . . 204-210 

CHAPTER VI 

LEVELING 

Definitions . . . . . .211 

Effect of Curvature 212 

Leveling considered under Three Heads . . ■ . _, 213 

Differential Leveling 214-217 

Profile Leveling 218-221 

Establishing a Grade Line 222-223 

Topographical Leveling 224-225 

Grading, Volumes 226 

TABLES 

PAGE 

I. Mean Refraction 36 

II. Errors in Azimuth (by Solar Compass) for One Minute Error in 

Declination or Latitude 37 

III. Refraction Correction, Latitude 40° 39 

IV. Latitude Coefficients 42 

V. Daily Variation of the Needle . . - .75 

VI. Declination Formulae 76, 77 

VII. Declination Values and Annual Change . . . . . . . ' 78 

VIII. Azimuths of Polaris at Elongation 81 

IX. Local Mean Time of Culminations and Elongations of Polaris . . 82 

X. Pole Distance (90° - Declination) of Polaris 88 

XL Linear, Surveyors', Square, and Cubic Measures, and the Metric System 147 

XII. Mensuration Formulae 150 

XIII. Trigonometric Formulae 151 

XIV. Squares, Cubes, Square Roots, and Cube Roots 152 

XV. Chords 153 

XVI. Stadia Tables . 155 

XVII. Logarithms of Numbers from 1 to 10,000 163 

XVIII. Logarithmic Sines, Cosines, Tangents, and Cotangents for Every Degree 

and Minute from 0° to 90° 179 

XIX. Natural Sines, Cosines, Tangents, and Cotangents for Every Degree and 

Minute from 0° to 90° . . . . . . . . .225 

XX. Auxiliary Table for Small Angles 241 

Appendix. Explanation of Logarithmic Tables ...,„., 243 



ELEMENTS OF 
PLANE SURVEYING 



PLANE SURVEYING 



INTRODUCTION 

1. Surveying is the art of making such measurements as will 
determine the relative positions of points on the earth's surface, both 
as to their distances, in a horizontal plane, from certain fixed refer- 
ence lines (that is, their geographical position) and their distances 
above, or below, a certain datum plane. It will be observed that the 
first determination fixes the vertical line through the point, the 
second giving its exact position in that vertical line.* 

2. Surveying has for its object the determination of the lengths 
and directions of lines on the earth's surface, the dividing and sub- 
dividing of tracts of land, the computation of areas and volumes, and 
the making of maps or plots to represent graphically portions of the 
earth's surface. 

3. In plane surveying, the surface of the earth (as given by 
the sea, for instance) is considered as a plane, the curvative being 

neglected. 

4. In geodetic surveying, or geodesy, the curvative of the earth 
is taken into account, and this must be done in extensive surveys. 

THE EARTH 

The few astronomical definitions here given will be followed by 
others as the student has occasion to use them. 

5. The axis of the earth is an imaginary line about which the 
earth rotates once in every 24 f hours. 

* The determination of the horizontal distances is commonly spoken of as plane 
surveying, while the determination of the vertical distances is called leveling. 
t See Art. 13. 

1 



2 PLANE SURVEYING [Art. 6 

6. The poles are the extremities of the axis, or the points where 
the axis meets the surface. 

Note. — The earth is not a perfect sphere, but is what is called an oblate spheroid, 
being flattened at the poles. According to "Clarke's Spheroid of 1866" (which is 
adopted by the United States Coast and Geodetic Survey) the dimensions of the earth 

are : Equatorial radius = 6,378,206.4 metres, 

= 3,963.307 miles. 
Polar radius = 6,356,583.8 metres, 

= 3,949.871 miles. 

7. The equator is that great circle of the earth which is midway 
between the poles. Its plane is perpendicular to the axis of the earth. 

8. Meridians are great circles passing through the poles perpen- 
dicular to the equator. 

9. Latitude and Longitude. — In geography, any place on the 
earth's surface is definitely known if its distance from the equator 
(that is, its latitude) and its distance from a certain reference 
meridian (that is, its longitude) are given. The English reckon 
longitude from the meridian through the Royal Observatory at 
Greenwich, while in the United States longitude is reckoned from 
the Washington meridian, or often from the Greenwich meridian. 

Latitude is north or south according as the place is north or south 
of the equator. 

Longitude is east or west according as the place is east or west of 
the reference meridian. 

Latitude is given in degrees, minutes, and seconds ; longitude, in 
degrees, minutes, and seconds, or hours, minutes, and seconds, one 
hour corresponding to 15°. 

10. A vertical line is the direction of gravity, as indicated by 
a plumb-line, at any point on the earth's surface. 

A horizontal line is a line perpendicular to a vertical line. 

11. A vertical plane is any plane that contains a vertical line. 
A horizontal plane is a plane perpendicular to a vertical line. 

12. A vertical angle is an angle whose sides lie in a vertical plane. 
A horizontal angle is an angle whose sides lie in a horizontal plane. 
An angle of elevation is a vertical angle, the lower side of which 

is horizontal, the other oblique. 

An angle of depression is a vertical angle, the upper side of which 
is horizontal, the other oblique. 



Art. 23] INTRODUCTION 3 

THE CELESTIAL SPHERE* 

13. The Celestial Sphere. — An observer on the earth is seemingly 
at the centre of a huge sphere, one half of which is the sky above 
him. This sphere, on the surface of which all the heavenly bodies 
appear to be, is called the celestial sphere. Owing to the rotation 
of the earth from west to east, this sphere, with its countless stars, 
apparently turns completely around from east to west once in about 
24 hours (really in 23 hr. 56 min. 4.1 sec. of ordinary time). 

14. The poles of the celestial sphere are the points where the axis 
of the earth produced pierces the celestial sphere. 

15. The zenith is that point of the celestial sphere directly over 
the head of the observer; the nadir is the opposite point directly 
under his feet. 

16. The meridian is the great circle passing through the pole 
and the zenith. 

17. The horizon is a great circle which is everywhere 90° from 
the zenith. 

18. Vertical circles are great circles passing through the zenith 
and perpendicular to the horizon. 

19. The altitude of a starf is its distance from the horizon 
measured on the vertical circle through the star. The zenith dis- 
tance is the complement of the altitude. 

20. The azimuth is the angle at the zenith between the meridian 
and the vertical circle which passes through the star. 

Note. — If a star's altitude and azimuth are given, its position in the heavens is 
known. 

21. The celestial equator is the intersection of the plane of the 
earth's equator with the celestial sphere. 

22. Hour-circles are great circles passing through the north and 
south poles of the heavens and perpendicular to the celestial equator. 

23. The North Pole. — In the apparent diurnal revolution of the 
stars, those near the north pole never set to an observer in the 

* The student will have occasion to use these definitions when he reaches the study 
of the theory of the solar compass, and in the chapter on the determination of a true 
meridian line. t Any heavenly body. 



4 PLANE SURVEYING [Art. 23 

northern hemisphere, but revolve in their so-called diurnal circles 
once a day* (see Art. 13). The north pole is conveniently marked 
by the pole-star (Polaris'), which is now only about 1J° from the 
pole. The pole-star may easily be found from its being nearly in a 
line with the so-called " pointers," two stars in the " dipper " (of the 
constellation of Ursa Major) (see Fig. 49). 

The pole is very nearly on a line joining Polaris with the star % 
Ursa Major, at the bend of the handle of the "dipper." This fact 
furnishes one method of getting a meridian line (see Art. 146). 

24. Culmination. — The circumpolar stars pass the meridian twice 
in 24 hours (see Art. 13), once above and once below the pole. These 
meridian passages are called respectively the upper and lower cul- 
minations. 

25. The Ecliptic. — The ecliptic is the path on the celestial sphere 
described by the sun in its apparent eastward motion among the 
stars that causes it to complete the circuit of the heavens once a 
year. The ecliptic is a great circle inclined to the equator at an 
angle of about 23J°. The poles of the ecliptic are obviously 90° dis- 
tant from the ecliptic. 

26. The Equinoxes. — The equinoctial points are the two points in 
which the ecliptic intersects the celestial equator. 

The vernal equinox is that one of these points which the sun 
passes in the spring, and the autumnal equinox is that passed in the 
autumn. 

27. The declination of a star is its distance north or south of 
the celestial equator measured on an hour-circle. Its north polar 
distance is the complement of its declination. 

28. The right ascension of a star is its distance measured on the 
celestial equator from the vernal equinox to that hour-circle which 
passes through the star. 

29. The hour-angle is the angle between the meridian of the 
observer's place and the star's hour-circle. 

30. The position of a star in the celestial sphere is determined if its 
right ascension and declination (or north polar distance) are given ; 
or if its hour-angle and declination (or north polar distance) are 
given. 

* Such stars are called circumpolar. 



Art. 31] 



INTRODUCTION 

z 




31. Figure 1 is supposed to represent the celestial sphere, the 
full circle being the meridian of the observer, and S being the posi- 
tion of a star, or other heavenly body. Then : 



HPZRN 
Z 

N 
HAR 



meridian, 

zenith, 

nadir, 

the horizon. 



EQ = the equator, 

BC — the ecliptic, 

PV, PD — hour-circles, 

ZA = a vertical circle, 
P = north pole, 



P' = south pole, 
V = vernal equinox, 
SA = altitude of the star, 
RA = azimuth of the star, 
ZS = zenith distance of the star, 
SD = declination of the star, 
PS = north polar distance, 
VD = right ascension of the star. 
ZPS — hour-angle. 



CHAPTER I 

INSTRUMENTS -THEIR USE AND ADJUSTMENTS 

32. Gunter's Chain. — Gunter's chain, so called after the inventor, 
is 66 ft. (4 rd.) in length, and is divided into 100 links, each link 
being 7.92 in. long.* 

A " link " consists of a straight piece (or link) of wire bent into 
rings at the ends, together with one or one and a half rings at each 
end according as there are two or three rings connecting the wires. 
The object of the connecting rings is to give flexibility to the chain. 
The best chains are made of steel wire and have a brass handle at 
the ends. Every tenth link is marked by a brass tag, the number 
of points on the tag, from each end to the centre, indicating the 
number of tenths from the end, the middle tag being oval. The 
"length" of the chain includes the handles. Gunter's chain is used 
only in obtaining the area of land when the acre is the unit. The 
fact that 10 square chains make an acre renders it a very convenient 
linear unit, though we often find distances expressed in terms of the 
smaller unit, the rod (or pole). In "rough or mountainous country 
it is customary to use a half-chain, made just like Gunter's, except 
that it is 33 ft. long and has only 50 "links." 

33. The Engineer's Chain is 100 ft. long and is divided into 100 
"links," each 1 ft. long. It is used in surveying railroads, high- 
ways, etc., and is now often used instead of Gunter's chain in ordi- 
nary land surveys. 

34. Tapes. — The steel tape, which is made in various lengths and 
is graduated into feet and inches, or feet and tenths, consists of a 
continuous piece of steel, about a quarter of an inch wide and ^^ of 
an inch thick. Instrument manufacturers make the tape in lengths 
ranging from 25 ft. up to 1000 ft., those in common use being 50 ft. 
or 100 ft. The longer tapes are used in bridge work or other special 
surveys. The steel tape has for many purposes taken the place of 
the engineer's chain. The linen tape is useful for domestic purposes, 

* See Table XI, page 147. 
6 



Art. 36] INSTRUMENTS — THEIR USE AND ADJUSTMENTS 7 

fencing, etc.; but owing to the ease with which it stretches and 
wears, it is too unreliable for the surveyor's use. 

Tapes sold under the name of metallic tapes are made of linen 
with fine brass wires woven throughout their lengths. While much 
better than the ordinary linen tape, they fall far short of steel tapes 
in efficiency and reliability. 

35. Standards. — To insure accuracy of measurement, all tapes 
and chains should be compared with the United States standard. 
For a small fee the United States Coast and Geodetic Survey, 
Washington, D.C., will test any chain or tape sent to them. The 
surveyor is advised to lay off and carefully mark the length of a 
chain that has been so tested on a curbing, or on the ground, mark- 
ing the ends by a line cut in a rock that has been securely bedded 
in the ground. He has then a ready means of testing his chain 
from time to time, a precaution which should be taken frequently. 

The ordinary Gunter's chain has 600 wearing surfaces. If each 
of these surfaces wears .005 of an inch, the chain will be lengthened 
three inches, so that in running a line a half a mile long an error of 
10 ft. will be made. The steel tape, while not so subject to change 
as the chain, is variable, owing to its contraction and expansion 
caused by changes in temperature. It is of standard length only for 
a given temperature (usually taken at 62° F.). 

It is beyond the scope of this work to describe instruments for 
precise measurement, or to explain the methods employed when great 
precision is required, as, for instance, in establishing a United States 
Coast and Geodetic Survey base line. 

36. Comparison of the Chain and Tape. — Some of the defects of 
the chain are : the number of wearing surfaces and consequent 
liability to change in length ; its weight, preventing its being kept 
horizontal unless resting on the ground ; when used in wet places the 
numerous rings are apt to become clogged with mud and grass ; the 
links are liable to be bent, and when straightened again they are 
always longer than before being bent. 

Among the advantages of the chain over the tape may be men- 
tioned its durability and, owing to its flexibility and strength, the 
ease with which it may be carried through brush and over rough 
ground. In rough country, for purposes not requiring great 
accuracy, the chain is much more satisfactory than the tape. 

The defects of the tape are its liability to break and the difficulty 
of repairing it. It is apt to rust, and thus the graduations are made 



8 PLANE SURVEYING [Art. 36 

indistinct. In a high wind it is inconvenient to use. As we have 
seen, it expands and contracts with changes of temperature (though 
the chain is of course similarly affected). 

Notwithstanding these obvious defects, the steel tape (if proper 
corrections are made for changes of temperature, etc.) is a far more 
reliable and convenient measuring instrument than the chain, and 
should always be used in preference to the chain when accuracy is 
desired. 

37. Pins. — For marking chain lengths, iron or steel pins, usually 
about 14 in. long, and bent in the form of a ring at the top, are used. 
Eleven is the most convenient number, though some surveyors use 
ten. They should be made heavier toward the bottom, so that when 
dropped from the hand they will fall plumb. 

38. Range Poles. — To range out a line, at least two poles are 
required. Range poles are usually of wood, 6 to 8 ft. long, painted 
in strips alternately red and white, and shod with a pointed iron 
shoe. An iron pipe is sometimes used, but is objectionable for 
several reasons. It is heavy, and if bent is hard to get perfectly 
straight again. When iron poles are used in compass surveying, 
great care must be taken not to leave them standing in the ground 
near enough to the instrument to cause a deflection of the needle. 

USE OF THE CHAIN (OR TAPE) 

39. To measure a Line with a Chain. — All distances should be 
measured horizontally. This method of measuring is prescribed 
by law in most states. It is the scientific way, for horizontal meas- 
urement is necessary in order to obtain a correct plot or map. 
What is really measured is the projection of surface lines on a 
horizontal plane ; and the area obtained is the area of the horizontal 
projection of the surface. Hence the chainmen must keep the chain 
horizontal by elevating the rear end in going uphill and the front 
end in going downhill. If the hill is very steep, it may not be 
possible to elevate the end of the chain enough to keep it horizontal. 
In this case it is customary to use a part of the chain at a time. 
Care must be taken to put the elevated end exactly over the pin. 
The safest way is to use a plumb-line, or a straight rod with a level 
attached to it.* 

* In the absence of a plumb-line or rod with level, the pin (see Art. 37) should 
be held loosely by the ring and, when steady, dropped, thus determining the point 
under the end of the chain. 



Art. 39] INSTRUMENTS — THEIR USE AND ADJUSTMENTS 9 

The following is the usual order of procedure : 

"Where eleven pins are used, the leader (front chainman) takes 
ten pins, leaving one with the follower (back chainman) at the 
beginning of the course. After the leader has the chain stretched 
out approximately along the course, the follower directs him by 
motioning with his hand, or by calling "right," "left," until he put.; 
him on a line with a range pole at the end of the course, or at 
some convenient place on the course. The leader then pulls the 
chain taut, being careful to keep it horizontal, and puts a pin on the 
line exactly at the end of the chain. The best way to do this is to 
take one pin in the (right) hand that grasps the handle of the chain, 
and, standing to one side so as not to obstruct the view, let the fol- 
lower line him with the pole. The leader, being in a stooping posi- 
tion, can readily stretch the chain by placing his right elbow on the 
inside of his right knee, using the latter as a fulcrum. When the 
follower calls " stick," he firmly presses the pin into the ground and 
says "stuck." The follower should in no case take his pin up till 
the leader signifies that he is ready by calling " stuck " (or " down "). 
When he has done this, the follower takes up the pin that has been 
left with him, and they both move on. The follower should warn 
the leader in some way that he is nearing another pin, and not stop 
him with a sudden and violent jerk on the chain. The follower 
holds the end of the chain at the pin, and as before " lines " the 
leader, calling "stick" when he is in the right position, and again, 
when the leader has stretched the chain and placed his pin, he calls 
" stuck," and the follower takes up his pin ; they move on, and 
repeat the process until the end of the course is reached. Should 
the distance exceed ten chains, when the leader has stuck his last 
pin, he calls " out," and the follower, dropping his end of the chain, 
walks up to the leader and gives him all the pins in his hand (ten in 
number). To avoid mistakes, both the leader and the follower 
should count the pins so as to make sure that ten have changed 
hands. Notice that the eleventh pin is still in the ground and is 
not counted, and that the distance up to this pin is ten chains. 
The leader moves forward and the measurement proceeds as before. 
When ten chains have thus been measured, there is said to be an 
" out," and the number of outs must be carefully kept. To assist 
the memory, a contrivance called an out-keeper is put on the plate 
of the compass (see Fig. 2). 

At the termination of a course, the end of the chain is held at 
the terminal stake, and the number of links back to the last pin is 



10 . PLANE SURVEYING [Art. 39 

counted. Notice that this last pin, within less than a full chain of the 
terminal stake, is never counted. 

Note. — Avoid the common mistake of reading from the wrong end of the chain, 
counting 40 links instead of 60, or of counting on the wrong side of a tag, putting 28 
for 32, for example. 

40. Too great emphasis cannot be laid upon the importance of 
careful chaining if the best results are desired. In much work the 
surveyor must depend upon green hands to " carry " the chain, but 
he should in such cases give them instruction as to the proper way 
to handle the chain, and he should in person watch them for a few 
courses (and every now and then later on) to insure especially, first, 
that the chain is held taut and horizontal ; second, that no gaps or 
overlaps are allowed when the pin is stuck, an error that is apt to 
occur on hilly ground ; third, that the number of links at the end 
of the course is properly counted. Some errors are compensating, 
but, as a rule, the fewer the errors, the more accurate the work. 

A compensating error is one that is as likely to be plus as minus, 
such errors tending to balance one another. A cumulative error is 
one that is constantly of the same sign; such errors, instead of 
tending to annul one another, accumulate as the work progresses. 
Thus, in chaining, the error in setting the piu is compensating, while 
an error due to erroneous length of the chain is cumulative. 

41. Error in Length of the Chain. — It is evident that if the chain 
is too long the number expressing the length of the line is too small, 
or the line is longer than the measurement makes it ; and if the chain 
is too short, the length of the line as found is too great. In either 
case, #, the true length of the line, is given by the following 
proportion : 

Length of standard chain : length of chain used 

= the distance measured : x. 

For example, with a chain one link too long the length of a line 
is 44.32 chains ; what is its true length ? 

Here 1 : 1.01 = 44.32 : x, or x = 44. T6 chains. 

42. Correction for Area. — As areas of similar figures are propor- 
tional to the squares of homologous sides, the formula for correcting 
area is : 

True area : computed area = the square of the length of chain used 
: the square of the length of standard chain. 



Art. 43] INSTRUMENTS — THEIR USE AND ADJUSTMENTS 



11 



For example, suppose the chain used is one link too long, and the 
area of a certain field is found by computation to be 544 square chains, 
what is the value of its true area, S ? 

Here S : 544 = (1.01) 2 : l 2 , 

or S — 554.93 square chains, true area. 



THE SURVEYOR'S COMPASS 

43. The Surveyor's Compass, a cut of which is shown below, 
consists of a plate of metal supporting at each end a standard, 
or sight-vane, perpendicular to the plane of the plate, the com- 
pass-circle, in which the magnetic needle swings, being at the centre 
of this plate. The compass-circle, which has a glass cover, is gradu- 
ated to half degrees and is figured from 0° to 90° each way from the 




Fig. 2. — Surveyor's Vernier Compass 

north and south points of the line of zeros. The four principal 
points are lettered clockwise* N.W. S.E., instead of N. E. S.W. 
as on a map, the reason for which will appear later on. The bubbles 
(or spirit-levels) are placed at right angles to each other, so as to 
level the plate in all directions. The standards have fine slits cut 
through nearly their whole length, terminated at intervals by circu- 
lar apertures, through which the object sighted upon is more readily 

* That is, in the direction of the hands of a clock. 



12 



PLANE SURVEYING 



[Art. 43 



found. The needle is essentially a magnetized bar of iron so suspended 
as to swing freely in a horizontal direction and settle in the magnetic 
meridian. It has an agate or jewelled centre which rests upon a sharp 
pivot at the centre of the compass-circle. As the precision of the 
needle depends not only upon its magnetic strength, but upon the 
ease with which it swings, it is very important to have the pivot 
sharp and smooth in the beginning and to keep it so. 

The north end of the needle is usually designated by a scallop or 
other mark, and the south end has a small coil of brass wire,* easily 




Fig. 3. — Compass with Telescope 



moved, so as to bring both ends to the same level. There is a lever 
worked by a screw, by means of which the needle, when not in use, 
can be raised up against the glass cover, thus preventing unnecessary 
wear on the pivot. 

The compass is usually fitted to a spindle, which is fastened to 
the head of the tripod or Jacob's staff by means of a ball-and-socket 
joint, a device which enables the surveyor to level the instrument 
very quickly. When a tripod is used, a plumb-bob is suspended 
immediately under the centre of the spindle. 

* To counteract the effect of the " dip " of the needle. 



Art. 46] INSTRUMENTS — THEIR USE AND ADJUSTMENTS 13 

44. In place of the sight-vanes a telescopic sight is often used. 
There is great advantage in this improvement, as it saves the eyes, 
and often enables the surveyor to set a pole at a greater distance, 
and through brush, where the pole could not be discovered with the 
naked eye. A good form * is shown in Fig. 3 on page 12. 

45. In the plain compass, an instrument used chiefly in the sur- 
vey of new lines where the variation f of the needle is not required, 
the compass-box is in the same piece with the main plate. 

The vernier compass, as illustrated on page 11, has its compass- 
circle, to which is attached a vernier, J movable about its centre a 
short distance in either direction, enabling the surveyor to set the 
zeros of the circle at any required angle with the line of sights. The 
number of degrees contained in this angle is read by the vernier. 
The superiority of the vernier over the plain compass consists in its 
adaptation to retracing the lines of an old survey, furnishing as it 
does a ready means of setting off the change in declination. 



VERNIERS ** 

46. A vernier is a short auxiliary scale, movable by the side of a 
longer scale called the limb, by means of which subdivisions of the 
limb may be measured. 

A division of the vernier is a little shorter or a little longer than 
a division of the limb, and it is this small difference that we are 
enabled to measure. 

A vernier is usually constructed by taking a length equal to a 
given number of spaces on the limb and dividing this length into 
a number of equal spaces, one more or one less than the number into 
which the same length on the limb is divided. If the number of 
spaces on the vernier is one more than the number of limb spaces 
covered, it is called a direct vernier ; if it is one less, a retrograde 
vernier ; because in the former case it is read in the direction of the 
motion, and in the latter case opposite to the motion. 

Figure 4 represents a direct vernier. Here the limb is supposed 
to be divided into feet and tenths of feet, and as shown by the first 
position of the vernier, AB, ten spaces on the vernier correspond to 

* Made by Young & Sons of Philadelphia. 
t " Change in the declination " ; see Art. 135. 
X See Art. 46. 
** Named after the inventor, Pierre Vernier. 



14 PLANE SURVEYING [Art. 46 

nine on the limb. Each space on the vernier is one-tenth less than 
a space (one-tenth of a foot) on the limb. The difference, then, 

4 5 6 



rrxi i ii i i i i i j 1 1 i i i i i i i i i i 

<V0 123456789 Vb/B A'\Q 12 3 4 5 6 7 8 9 10/B' 

Fig. 4. — Vernier Plate 

between a space on the limb and one on the vernier is one-tenth of 
one-tenth of a foot, or .01 of a foot. 

This difference is called the least count of the vernier, and is the 
smallest distance it can measure. 

47. To find the least count of any vernier, let 

s = length of one space on limb, 
v = length of one space on vernier, 
n = number of spaces on vernier ; 
then ns = nv ± s. 

For direct vernier we use the + sign, which gives 

s — v = - = least count. 
n 

48. In Fig. 4, with vernier at the position AB, the reading is 
even 4 ft. Now, if we suppose the vernier to slide along the limb 
toward the right till the vernier line 1 coincides with the division on 
the limb next after the 4-ft. mark, the vernier has evidently moved 
up .01 ft., and the reading is 4.01. If we move the vernier a little 
farther, so that the line 2 coincides with a line on the limb, the ver- 
nier has moved .02 ft. It will be noticed that the feet and tenths 
are given by the position of the zero-line of the vernier and are read 
upon the limb-scale, while the hundredths are given by that line 
of the vernier scale that coincides most nearly with some line of 
the limb-scale. In the second position of the vernier, A'B f in our 
figure, we see that the seventh line on vernier coincides with one 
above it, and the zero-line is between 5.2 and 5.3, giving the final 
reading 5.27. 

49. There are many different forms of verniers ; a very common 
form, shown in Fig. 5, is used when the main (limb) scale is graduated 



Art. 52] INSTRUMENTS — THEIR USE AND ADJUSTMENTS 



15 



into degrees and half degrees. Here 30 spaces on the vernier corre- 
spond to 29 on the limb, and the least count is -^ of 30 min. = 1 min., 
thus making it possible to read the angle to minutes. 




Fig. 5 

This double vernier (really two verniers, the zero points of which 
coincide) is very common on transits. The object of this double 
vernier is to avoid the inconvenience of reading backward when 
the motion of the limb is reversed. 

50. The principle of the vernier seems to be sufficiently explained 
in Art. 48. The student is advised to draw carefully on a sheet of 
paper a graduated scale, and to draw his vernier on the edge of a 
piece of cardboard. Then, by sliding this movable vernier along 
the main scale, he can practise reading any angle he pleases. On 
taking any instrument into his hands, he should first examine the 
vernier to ascertain its least count; that is, to learn what is the 
smallest reading that it will give. Should he be called upon to use 
a vernier with a peculiarity that he cannot master, let him write to 
the maker of the instrument for the desired information. 



ADJUSTMENTS OF THE COMPASS 

51. It is necessary for the surveyor to consider the adjust- 
ments of : 

(1) The bubbles. (2) The standards. (3) The needle. 

52. First, the Bubbles. — In attempting to " level " the compass, 
if it is found that the bubbles do not stay in the centre of the tube 
as the compass-plate is turned around, the compass is said to be out 
of adjustment, — the real difficulty being that the plane of the bub- 
bles is not parallel to the plate.* 

*This assumes that the plate has been made truly perpendicular to the spindle by 
the manufacturer. 



16 PLANE SURVEYING [Art. 52 

To adjust the Bubbles. — Set up the instrument and bring the 
bubbles to the middle of the bubble-tube by pressure of the hands 
on the plate. Turn the compass half-way around, and if the bubbles 
remain at the middle, no adjustment is needed. If they do not, 
bring each half-way back to the middle by means of the screws at 
the ends. Level the plate'again, and repeat the first operation until 
the bubbles will remain in the middle during an entire revolution 
of the plate. 

53. Second, the Standards. — Observe through the slits a fine 
plumb-line, and if either sight fails to range with it, that sight must 
be adjusted by riling its under surface on the side that seems the 
highest. 

54. Third, the Needle. — This adjustment is needed if the needle 
will not in any position cut opposite degrees. Having the eye 
nearly in the same plane with the graduated circle, with a small 
splinter of wood bring one end of the needle in line with any promi- 
nent graduation of the circle, as the zero, and notice whether the 
other end corresponds with the degree on the opposite side. If it 
does " cut " opposite degrees in this and any other position, the 
needle is in adjustment. If not, bend the centre pin (pivot) by 
applying a small wrench about one-eighth of an inch below the point 
of the pin until the ends of the needle are brought into line with the 
opposite degrees. Then, holding the needle in the same position, 
turn the plate half-way around, and note whether it now cuts oppo- 
site degrees ; if not, correct half the error by bending the needle, 
and the remainder by bending the pivot. This operation should be 
repeated until perfect reversion is obtained. Then try again on 
another part of the circle, and, if any error appears, correct by bend- 
ing the pivot only, the needle being already made straight b}^ the 
previous operation. 

55. Only the beginner in the study or practice of surveying could 
ask the question, Why cannot instruments be made so that no adjust- 
ment would be necessary ? A partial answer to such a question, 
however, may not be out of place. One example will suffice : sup- 
pose the bubbles were rigidly secured to the plate and made exactly 
parallel to the plate (perpendicular to the vertical axis of the instru- 
ment) by the manufacturer. This essential relation may be disturbed 
in many ways. Even if the plate, or the level tube case, has not been 
bent by a fall or sudden jar, as may readily happen, changes of tem- 
perature, causing uneven expansion and contraction, or the ordinary 



Art. 59] INSTRUMENTS — THEIR USE AND ADJUSTMENTS 17 

wearing of the axis, would soon throw the plane of the bubbles out of 
its proper position, and then in order to adjust the parts the instru- 
ment would have to be sent to a repair shop, involving expense and 
delay. Thus we see the wisdom of making those parts of all instru- 
ments liable to such derangement movable (or adjustable). 

This all-important matter is always troublesome to the beginner ; 
but he should be diligent in acquainting himself with the reason for 
every step, so that he can make each adjustment intelligently, and 
not in a purely mechanical way. The surveyor should test his in- 
strument frequently. If it needs adjusting, there will usually be no 
practical difficulty in correcting the error, provided that the instru- 
ment has been firmly set and the operator handles the parts carefully. 
Every movement of the hand should be deliberate (not slow), and 
a jerky motion is always to be avoided. 

THE USE OF THE COMPASS 

56. True Meridian. — A meridian plane is any plane passing 
through the axis of the earth, and its intersection with the surface 
is a meridian (compare Art. 8). 

57. Magnetic Meridian. — If a magnetic needle is suspended freely 
(as should be the case in the needle-compass) and allowed to come 
to rest, it will point toward the north magnetic pole,* and the inter- 
section of the vertical plane containing the needle with the surface of 
the earth is called the magnetic meridian. Meridian lines converge 
toward the poles, but in limited surveys (such as ordinary farms) 
they may be considered parallel without appreciable error. 

58. A course is a line measured on the ground. The bearing of a 
course is the angle which the course makes 
with the meridian. 

59. The needle-compass is used to get 
the magnetic bearing of a line, or course. 
For example, to get the bearing of a 
course AB, set up the compass over the 
point A, level the plate, and sight a pole 
held at B ; that is, put the north and south 
(N. and S.) line of the compass-box in 
line with AB, the south end being toward 

* See Chapter III (ii). 




18 PLANE SURVEYING [Art. 59 

the observer.* After the needle has come to rest, we read the angle 
(less than 90°) which it makes with the N. and S. line of the compass- 
box (in this case, 48°) ; the bearing is then N. 48° E. In our diagram 
PP' represents the position of the needle ; that is, the magnetic merid- 
ian. We can now understand why the E. and W. points on the 
compass-circle are reversed. While the zero (N. and S.) line of the 
compass-circle points toward B, the needle points toward P, and 
the course being east of the needle, the needle is west of the course 
(or the zero line of the compass-circle). Were the circle lettered 
like the points of the compass on maps N. E. S. W. (clockwise), the 
surveyor would first read this bearing N. 48° W., and would then have 
to change it mentally to N. 48° E. It is to avoid the necessity of 
this mental change that the circle is lettered N. W. S. E. (clock- 
wise). 

If the bearing of AC was sought, the angle PAO being 40°, 
the reading would be N. 40° W. If P 1 AD = 15°, the bearing 
of AD is S. 15° W., and if P'AE=$0% the bearing of AE is 
S. 80° E. Special field operations with the compass will be de- 
scribed in Chapter III. 

THE SOLAR COMPASS 

60. We give below a cut of Burt's solar compass, an instrument 
by means of which the bearing of a course with the true meridian is 
obtained by observations on the sun. The theory and use of this 
instrument will be discussed under the head of the Solar Attachment 
to the Transit (see Art. 76). 

THE TRANSIT t 

61. The transit was invented by Messrs. J. W. Young & Sons,J 
instrument-makers, of Philadelphia, in 1831. It differs from the 
theodolite, an instrument much used by English engineers, in that 
the telescope is constructed so that it can make a complete revolu- 
tion on its horizontal axis. It is the most important of all the 
engineering instruments. Its immense value is due to the telescope, 
which gives precision in sighting, and the graduated circle, by which 
angles can be read with ease and accuracy. All the other parts 

* The man with the instrument is often spoken of as the observer or operator, 
t Often spoken of as the Surveyor's or Engineer's Transit, to distinguish it from an 
astronomical transit. 

| Really by William J. Young, of whom J. W. Young & Sons are the successors. 



Art. 62] INSTRUMENTS — THEIR USE AND ADJUSTMENTS 



19 



facilitate the use of these. The essential parts, as shown in our cut, 
are the telescope with its axis and two standards, the circular plates 
with their attachments, the sockets upon which the plates revolve, 
the leveling-head, and the tripod upon which the whole instrument 
stands. 




Fig. 7. — Solar Compass 

62. The telescope,* which is usually from 10 to 11 in. long, is 
firmly secured to an axis, having its bearings fitted in the standards, 
which are high enough to allow the telescope to be turned completely 
over on its axis. A skeleton view of the telescope is given in 

* For a more detailed description of a telescope, the student is advised to consult 
''Engineers 1 Surveying Instruments," by Ira O. Baker, or any good work on Physics. 



20 



PLANE SURVEYING 



[Art. 62 



jS 




Fig. 8. The object-glass is a compound, achromatic lens, and is 

placed at the end of a slide having two bearings, one at the end of 

the outer tube, the other in the ring CO suspended within the tube 

by four screws, only two of which are shown in the cut. Both the 

object-glass and eye-piece are moved in and 

|>^CZ!lilMf ou ^ by pinions, and are thus adjusted to 

<j u_^ z*ij l ^^ proper focus. The eye-piece is made 

L J\ up of four lenses, which form a compound 

H jr| microscope having its focus in the plane 

of the cross-wire ring, BB. Sometimes an 
eye-piece with two lenses is employed ; 
this arrangement gives an inverted image 
of the object seen, which is considered a 
disadvantage, and is seldom used by American 
engineers. It has, however, the decided ad- 
vantage of giving more light, and should be 
employed in all high-grade instruments. The 
object-glass collects the rays of light which 
come from an object, converges them to a 
focus at the cross-wires, and there forms a 
minute, bright image, which the eye-piece, 
acting as a microscope, magnifies and conveys 
to the eye. 

The cross-wires are two wires of very fine 
platinum cemented into cuts on the surface 
of a metal ring called the reticule. Spider 
webs have been the favorite material for these 
wires, but platinum wires are rapidly replac- 
ing them. These wires are placed at right 
angles to each other, so as to divide the open 
space in the centre into quadrants. The 
reticule is held in place by four screws, two 
of which are shown in the cut at BB. 

The line of collimation, or line of sight 
when the instrument is in adjustment, is the 
imaginary line passing through the intersec- 
tion of the cross-wires and the optical centre 
of the object-glass. It is important that one of the wires should 
be exactly vertical, the other horizontal, when the instrument is 
set up and leveled ; the vertical wire being used in measuring 
horizontal angles, and the horizontal in measuring vertical angles 





Fig. 8 



Art. 63] INSTRUMENTS — THEIR USE AND ADJUSTMENTS 



21 



or in using the instrument as a level. Many transits have two 
additional horizontal wires for stadia work, the use of which will 
be explained later. 




Fig. 9. — Surveyor's Transit 



63. In the centre of the upper plate, between the standards, there 
is a compass-circle, with magnetic needle, which does not materially 
differ from that on the ordinary needle-compass. It may or may 
not have a vernier for setting off the variation of the needle. The 



22 



PLANE SURVEYING 



[Art. 63 



cut on page 21 represents the surveyor's transit as made by W. & 
L. E. Gurley, and below is given a sectional view of this instru- 
ment, showing the interior construction of the limbs, spindles, 
sockets, etc. 

The transit proper, which rests upon the leveling-head rigidly 
attached to a tripod, consists of two plates, called the upper and 
lower plates, the former carrying the compass-circle, standards, etc., 
and the latter the graduated circle, which is read to minutes by 
means of a vernier on the upper plate. In this book we shall adopt 
the usage of calling the upper plate the alidade, the lower plate, the 
limb.* These plates have two concentric vertical axes. The alidade 




Fig. 10. — Sectional View of Transit 



and limb may be firmly fastened together by a clamp, I)F in the sec- 
tional view, to which is attached a tangent screw for giving them a 
slow motion around each other. The limb has another clamp and 
tangent screw to fix it to its spindle and to give it a slow motion 
around the spindle. When the plates are clamped together, the instru- 
ment turns as a whole about the spindle of the limb. The verniers 
FT^are attached to the alidade diametrically opposite to each other. 
One vernier acts as a check upon the other. The alidade also carries 
the two bubbles, placed at right angles to each other, so that the 
instrument may be leveled in all directions. The limb (at B in sec- 

* Sometimes called the horizontal limb, to distinguish it from the vertical limb 
for circle) attached to the axis of the telescope. 



Art. 66] INSTRUMENTS — THEIR USE AND ADJUSTMENTS 23 

tional view) is usually divided on its upper surface into degrees and 
half-degrees and figured in two rows from 0° to 360°, or from 0° to 
90° each way. 

The leveling -head consists of two plates connected in such a 
way that they are inclined to each other or made parallel at will 
by the four leveling-screws. The lower of these plates usually 
has a " shifting centre," which facilitates setting the plummet pre- 
cisely over a point. The leveling-head is firmly screwed to the 
bronze head of the tripod. 

In addition to the parts mentioned above, it is convenient, and 
often indispensable, to have a graduated vertical circle (as shown in 
our cut), or arc, attached to the axis of the telescope, and a bubble 
parallel to the telescope to which it is fastened. The telescope-bub- 
ble enables the surveyor to do leveling work with the transit, and 
both it and the vertical circle (or arc) are necessary for stadia work 
and for obtaining angles of elevation. When the bubble is in the 
centre, the vernier of the vertical arc should read zero. 

ADJUSTMENTS OF THE TRANSIT 

64. The four principal adjustments of the transit have to do with: 

(1) The plate-bubbles. 

(2) The line of collimation. 

(3) The standards. 

(4) The telescope-bubble. 

65. First. — To make the Plane of the Plate-bubbles Perpendicu- 
lar to the Vertical Axis. This adjustment is the same as for the 
needle-compass (see Art. 52). 

66. Second. — To make the Line of Sight* Perpendicular to the 
Horizontal Axis of the Telescope. 

When this is done, the line of sight will generate a plane when 
the telescope is revolved about its horizontal axis. 

On tolerably level ground, having set up the transit and leveled 
it, sight a point (marked by a tack in a stake) 200 to 300 ft. distant, 
and clamp both the alidade and the limb ; revolve the telescope 
about its horizontal axis, and mark a point at the same distance 
from the instrument in the opposite direction. Then loosen the 
upper clamp, and turn the instrument about its vertical axis until 
the first point is in line again, clamping the plates firmly together 

* After adjustment the line of sight is the line of collimation (see Art. 62). 



24 



PLANE SURVEYING 



[Art. 66 



when this is done. Revolve the telescope about its horizontal axis, 
and sight toward the second point. If this is found to be in line, 
the transit is in adjustment, and is exactly on the straight line 
between the two points. If not, the intersection of the cross-wires 
should be brought back one-fourth of the distance to this second 
point by means of the pair of reticule screws on the sides of the 
telescope which move the vertical wire. This is done by loosening 
one screw and tightening the other. 

Test by repetition, and, if need be, repeat the process till the 
adjustment appears perfect. The following explanation taken from 
Professor Lane's little book on adjustments * will make this process 
clear. 

67. It is convenient here to use the following abbreviations : 

A, horizontal axis of telescope. 
(7, line of collimation. 
I, intersection of A and O. 
S, axis of spindle. 

Subscripts to A and indicate their positions during different 
E q stages of the work. 

" Assume a hori- 
zontal circle (Fig. 11) 
with centre i", and let 
C x differ a from IE, a 
perpendicular to A. 
Revolving C x about A x 
till it again cuts the 
circle, it takes the posi- 
tion (7 2 , differing a from 
IB, and therefore 180° 
— 2 a from C v Turn- 
ing the instrument 
about S 180° -2 a, C 2 
becomes C 3 in coinci- 
dence with C v while 
A 2 takes the position A s . 
Revolving C 3 about A z 
until it again cuts the 
circle, it takes the position (7 4 , differing a from C 5 perpendicular to 




* "Adjustments of the Compass, Transit, and Level," by A. V. Lane (Ginn & Co.). 



Art. 69] INSTRUMENTS — THEIR USE AND ADJUSTMENTS 25 

A 5 . 5 and ID, respectively perpendicular to A 5 and A 2 , differ by 
the same angle as those positions of the axis ; viz. 2 a. Since the arc 
45 subtends a, & D subtends 2 a, and D 2 subtends a, the ratio of the 
arc 45 to the arc 42 is clearly that of a to 4 a, or one-fourth. If then 
(7 4 were at <7 5 , one-fourth of the way toward <7 2 , it would be per- 
pendicular to the horizontal axis of the telescope, and hence in 
adjustment." 

Note. — Here and elsewhere when the instrument is said to be set up, it is 
understood that it is properly set up; that is, the legs are firmly planted in the 
ground, the plate (before using the leveling-screws) is reasonably level, and the 
plummet, when a station is occupied, is exactly over the point. The surveyor 
should see that no bystander puts his foot near a leg of the tripod, for by so doing 
he may readily throw the instrument out of its true position, especially in soft 
ground. See Art. 55 for other suggestions. There is more than one method for 
nearly all of the adjustments ; the author gives the one that he prefers. 

68. Third. — To make the Horizontal Axis of the Telescope Per- 
pendicular to the Axis of the Instrument. 

When this is done, the line of sight will generate a vertical 
plane as the telescope is revolved. To secure this result, the stand- 
ards must be exactly of the same height ; hence we have called this 
the standard adjustment. On nearly level ground in front of a 
high building, set up the transit at a distance about equal to the 
height of the building. Sight * a point (it is usually easy to find a 
well-defined mark that will serve the purpose) near the top of the 
wall, and, having clamped both the alidade and limb, lower the 
telescope till it is about horizontal ; then find or mark a point in 
the line of sight near the bottom of the wall. Unclamp the plates, 
revolve the telescope about 180° on both the vertical and hori- 
zontal axes, and again sight the upper point. Clamp the plates and 
lower the telescope to the lower point. If the line of sight cuts this 
point, the standards are in adjustment. If not, correct one-half 
the difference by raising or lowering the adjustable end of the axis. 
Test by repeating. 

69. Fourth. — To make the Axis of the Telescope-bubble Parallel 
to the Line of Sight. 

This adjustment is necessary only if the instrument is to be used 
as a level. Here we use the so-called "peg-adjustment," as follows : 
On a tolerably level piece of ground drive two pegs firmly into the 

*Here and in other places the verb "sight" will be used as an abbreviation for 
" get the line of sight upon." 



26 PLANE SURVEYING [Art. 69 

earth, about 200 or 300 feet apart. Set the instrument near one 
of them, say peg No. 1, so that when the leveling-rocl * is held 
vertically upon it, the eye end of the telescope will swing about half 
an inch from its face. Turn the eye end of the telescope toward 
the face of the rod, the bubble being in the middle of its tube ; look 
through the object end and set a pencil point on the rod at the 
centre of the small field of view. This pencil point gives the eleva- 
tion of the instrument, which we will call a. Now hold the same rod 
on peg No. 2. With the object end toward the rod, being careful 
that the bubble is still in the middle of the tube, set the target and 
call the reading b. Next move the instrument and set it up near 
peg No. 2 ; get the height of the instrument, a 1 ', as was done at 
peg No. 1, by sighting through the object end the rod held on peg 
No. 2. Then set the target, in usual way, on the same rod held 
on peg No. 1, and call this reading b'. 

Now, if the line of sight is parallel to the axis of the bubble, the 
difference between a and b must be equal to the difference between 
b' and a', because each of these differences represents the same thing ; 
viz., the difference of elevation of the two pegs. Hence, if 

a-b = V -a\ (1) 

no adjustment is necessary. If the instrument is not in adjustment, 

we have -, ,-,, ,, 7 ^ . 

a — b — (b' — a') = d, (2) 

or, a + a' -(b + b f )=d, (3) 

where d is twice the deviation of the line of sight from the axis of 
the bubble for the given distance ; (3) shows that the line of sight 
inclines down when d is positive, and up when d is negative. 

Correct the error by moving the target a distance equal to J d, 
up if d is plus, down if d is minus, get the line of sight by elevating 
or depressing the telescope, f and bring the telescope-bubble to the 
middle by means of the screw at the end of the bubble-tube ; or 
else leave the telescope undisturbed with the bubble in the middle, 
and adjust the line of sight to read upon the target by moving the 
reticule. Notice that when this is being done the instrument is at 
its second position, near peg No. 2, and the rod is being held on peg 
No. 1. Repeat the process till no error is apparent. 

The following example is given by way of illustration. 

* See Art. 96. 

f The simplest way of doing this with the transit is to revolve the telescope a little 
on its horizontal axis. If we were adjusting a Y- level, we should use the leveling-screws. 



Art. 72] INSTRUMENTS — THEIR USE AND ADJUSTMENTS 27 

70. Peg adjustment of level (or transit with telescope-bubble). 



a 




D 






E 









■ m 






A 




K_ 









Fig. 12 



AC=a =4.5, 
BD=b =5.5, 
BE=a' = 4,7, 
AF = V =2.6. 



( a + a F )-(b + b f )=d, 
(4.5 + 4. 7)- (5.5 + 2.6) = 
.:d = + 1.1. 



AB is the natural slope of the ground where adjustment is made, 
peg No. 1 being at A, peg No. 2 at B. 

AK is true horizontal or level line.* CD is line of sight in first 
position of the instrument, near A. 

EF is line of sight in second position of the instrument, near B. 

EH being drawn parallel to D (7, HF = d ; and if the target is 
moved to 0, distant ^ d above F, 0E will be parallel to AK. 

Hence, in this case, move the target up (because d is +) 0.55, 
and after getting the intersection of the cross-wires upon it, bring 
the bubble to the centre of its tube. 

71. Other adjustments of the transit are occasionally required, 
but as the construction of instruments has reached such perfection 
as to render these unnecessary in most cases, they will not be given 
here. 

THE USE OF THE TRANSIT 

72. To measure a Horizontal Angle. — Set up the transit with the 
plummet exactly over the vertex of the angle. With the alidade 
clamped at zero for convenience (any other gradua- 
tion point would answer) direct the telescope to 
some point on one side of the angle, as at C. A 
small tack driven into the top of a wooden peg is a 
convenient mark for a point. Then clamp the limb 
to the spindle and with the tangent screw get the line of sight 
exactly on the point. Then loosen the alidade and sight a point 

* Neglecting the curvature of the earth. 




Fig. 13 



28 PLANE SURVEYING [Art. 73 

on the other side of the angle, such as A; with the upper tangent 
screw bisect this point exactly and read the angle passed over from 
zero. This reading will be the value of the angle. As a check, and 
to counteract possible errors in graduation, both verniers should be 
read and the mean of the two readings taken if there is a slight 
difference. 

73. To measure a Vertical Angle. — As one- of its sides is usually 
horizontal, a vertical angle is generally an angle of elevation or 
depression (see Art. 12). If the vernier of the vertical arc reads 
zero when the telescope-bubble is in the centre of its tube, simply 
sight the point ; the reading of the vertical arc gives at once its 
elevation or depression. If neither side of the angle is horizontal, 
the algebraic difference of the elevation (or depression) of the two 
sides gives the value of the vertical angle. 

74. To produce a Straight Line with a Transit. — Knowing the 
direction that the line is to take from its initial point, A, set up the 
instrument over this point, and set another point, B, at a convenient 
distance along the line. Then move the transit and set it up over B. 

ABODE 

© o © o , © 

Fig. 14 

Sight the first point, J., then, having both plates clamped, revolve 
the transit on its horizontal axis and. fix another point, (7, which (if 
the instrument is in adjustment) will be on the prolongation of the 
line AB. Then move the instrument to (7, sight back to B, and, 
having the plates clamped, revolve the telescope as before and fix 
another point, D, at some convenient distance ahead. By continuing 
the process, a line may be prolonged indefinitely. 

Further use of the transit will be explained when we take up the 
subject of Transit Surveying. 

THE SOLAR ATTACHMENT 

75. The solar compass, an instrument contrived for determining 
a true meridian, w T as invented by William A. Burt, of Michigan, 
and patented by him in 1836. It came into general use in the sur- 
veys of the United States public lands. Its great superiority over 
the needle-compass lies in the fact that the bearings of lines are 
determined with reference to the true meridian and not to the vary- 
ing magnetic meridian. It will be seen from Fig. 7, page 19, that 



Art. 77] INSTRUMENTS — THEIR USE AND ADJUSTMENTS 29 

the solar apparatus takes the place of the needle. The work, how- 
ever, can be done with more ease and accuracy by the use of the 
transit with a solar attachment, and as the principle is the same, 
we shall not enter into any description of the solar compass. 

76. The solar attachment as made by Messrs. W. & L. E. Gurley 
(Fig. 15, page 30) is essentially the solar apparatus of Burt, placed 
upon the cross-bar of the ordinary transit, the polar axis being 
directed above instead of below as in the solar compass. This cut 
also serves to give a graphic illustration of the theory of the solar 
apparatus. It will be understood by reference to definitions, Arts. 
13 to 29. 

The polar axis of the attachment is connected by means of four 
screws with a disk, which is securely screwed to the telescope axis. 

The hour-circle surrounding the base of the polar axis is easily 
movable about it, and can be fastened at any point desired by two 
flat-head screws above. It is divided to 5' of time, is figured from 
I to XII, and is read by a small index fixed to the declination arc and 
moving with it. 

The declination arc has a radius of about five inches and is divided 
to quarter degrees. Its vernier, reading to half minutes, is fixed to 
a movable arm, at each end of which is a rectangular block of brass 
in which is set a small convex lens, having its focus on the silver 
plate A (see page 31) on the opposite block. The arc of the decli- 
nation limb is turned on its axis and one or the other solar lens used, 
as the sun is north or south of the equator ; the cut shows its posi- 
tion when the sun is north. 

Latitude Arc. — The latitude is set off by means of the large 
vertical limb of the transit, graduated from the centre each way. 
The usual tangent movement to the telescope axis serves to bring 
the vertical limb to the proper elevation. A telescope-bubble (on 
under side of the telescope) is indispensable in the use of the solar 
attachment. 

THEORY OF THE SOLAR ATTACHMENT 

77. In our diagram the circles shown are intended to represent 
those supposed to be drawn upon the concave surface of the heavens. 
Compare Fig. 1, page 5. When the telescope is set horizontal by 
its bubble, the hour-circle will be in the plane of the horizon, the 
polar axis will point to the zenith, and the zeros of the vertical arc 
and its vernier will coincide. Now, if we incline the telescope, 
directed north, as shown in the cut, the polar axis will descend, 



30 



PLANE SURVEYING 



[Art. 77 



from the direction of the zenith, through an angle, laid off on the 
vertical arc, equal to the co-latitude * of the place where the instru- 
ment is supposed to be used. Then, when the sun is on the equa- 
tor (at the vernal and autumnal equinox), the telescope and the 




Fig. 15. — Solar Attachment 

arm of the declination arc, if fixed at zero, will " follow " the path 
of the sun when the transit is revolved about its vertical axis. 
When, however, the sun passes above or below the equator, his 
declination f can be set off upon the arc, the arm carrying the lenses 

* That is 90° - the latitude. t Art. 27. 







Wt 



m^ A 



Art. 78] INSTRUMENTS — THEIR USE AND ADJUSTMENTS 31 

can be made to follow his path as before, and his image brought 
into proper position. 

In order to do this, it is necessary not only that the latitude and 
declination be correctly set off upon their respective arcs, but also 
that the instrument be moved in azimuth * until the polar axis 
points to the pole of the heavens, or in other words, is placed in the 
plane of the meridian ; and thus the position of the sun's image will 
indicate not only the latitude of the place, the declination of the sun 
for the given hour and the apparent time, but it will also determine 
the meridian or true north and south line passing through the place 
where the observation is made. 

The interval between the two equatorial lines, cc, as well as 
between the hour lines bb, is just sufficient to include the circular 
image of the sun, as formed by the solar lens on the opposite end of 
the revolving arm. When the solar attachment 
is accurately adjusted and the plates of the transit 
made perfectly horizontal, the latitude of the place 
and the declination of the sun for the given day and 

' . , nr , . & . J Fig. 16 

hour being also set oft on their respective arcs, 
and the instrument set approximately north by the magnetic needle, 
the image of the sun cannot be brought between the equatorial lines until 
the polar axis is placed in the plane of the meridian of the place, or 
in a position parallel with the axis of the earth. 

Thus we obtain for our reference line a true north and south line, 
for the slightest deviation from this position will cause the image to 
pass to one side or the other of the lines. 

The weak point of this form of attachment is that the naked 
eye must determine when the sun's image is exactly in position. 
The method of running lines with this attachment will be under- 
stood from the explanation of the mode of using the improved form 
of solar attachment described below. 

78. To Adjust the Solar Attachment. — The surveyor should first see that 
the transit itself is in perfect adjustment. There are then three principal adjust- 
ments of the attachment. 

First. — To adjust the Solar Lenses and Lines. Remove the declination arm and 
replace it by the adjuster, a short bar furnished with the instrument. Now place 
the declination arm upon the adjuster, turn one end to the sun, and bring it into 
such a position that the image of the sun is precisely between the equatorial lines 
on the. opposite plate. Turn the arm over (not end for end), and again observe 
the sun's image. If it remains between the lines as before, the arm is in adjust- 

* Revolved about its vertical axis. 



32 PLANE SURVEYING [Art. 78 

merit. If not, loosen the three small screws which hold it to the arm and move 
the silver plate under their heads until one-half the error is removed. Bring the 
image again between the lines, and repeat the operation as above on both ends of 
the arm, until the image will remain between the lines of the plate in both posi- 
tions of the arm, when it will be in adjustment, and the arm may be replaced in 
its former position on the attachment. 

Second. — To adjust the Vernier of the Declination Arc. Set the vernier at zero, 
and with one lens toward the sun bring his image exactly between the equatorial 
lines on opposite plate. Clamp the telescope axis and revolve the arm until the 
image appears on the other plate. If precisely between the lines, the adjustment 
is complete ; if not, move the declination arm till the image is centred, clamp the 
arm, and correct one-half the apparent index error by loosening the two screws 
that hold the vernier and moving the vernier. Test again. 

Third. — To adjust the Polar Axis. This consists in making the polar axis 
perpendicular to the axis of the telescope. Level the instrument with the plate- 
bubbles in connection with the long telescope-bubble, until the latter will appear in 
the middle of its tube during a complete revolution of the transit upon its spindle. 
Bring the declination arm of the solar apparatus in the same vertical plane 
with the telescope, place the adjusting level (which comes with the solar attach- 
ment) upon the top of the rectangular blocks, and bring the bubble of this level 
into the middle by the tangent screw of the declination arc. Then turn the arc 
half-way around, making it parallel to the telescope, and note the position of the 
bubble. If in the middle, the polar axis is vertical in that direction; if not, cor- 
rect one-half of the error by the adjusting screws under the base of the polar axis. 
Test by repeating the operation. Pursue the same course in adjusting the arc in 
the second position, over the telescope axis. When this is done, the bubble will 
remain in the middle during an entire revolution of the arc, and the adjustment 
is completed. This is the most delicate and important adjustment of the solar 
attachment- 
It is sometimes necessary to adjust the hour-arc. As the index of the hour-arc 
should read apparent time, the method of making this adjustment will readily 
suggest itself. 

THE SAEGMULLER SOLAR ATTACHMENT 

79. The Saegmuller solar attachment * seems to be superior to 
any form f yet devised. Fig. IT, page 33, shows a transit with the 
Saegmuller attachment, which can be put on any transit that pos- 
sesses a telescope-bubble and vertical circle. The improved attach- 
ment, as now made, is shown in Fig. 18. 

It consists essentially of a small telescope and level, the tele- 
scope being mounted in standards, in which it can be elevated or 
depressed. The standard revolves around an axis, called the polar 
axis, which is fastened to the telescope axis of the transit instrument. 

* Invented by G. N. Saegmuller, of Washington, D.C., in 1881, and manufactured 
by him and other instrument-makers. 

t A similar form is made by C. L. Berger & Sons, Boston, Mass. 



Art. 



INSTRUMENTS — THEIR USE AND ADJUSTMENTS 



33 



The telescope, called the " solar telescope," can thus be moved in 
altitude and azimuth. Two pointers attached to the telescope tc 




Fig. 17. — Transit with Saegmuller Solar Attachment 



34 



PLANE SURVEYING 



[Art. 80 



approximately set the instrument are so adjusted that when the 
shadow of the one is thrown on the other the sun will appear in 
the field of view. 

ADJUSTMENT OF THE APPARATUS 

80. First. — The transit must be in perfect adjustment, 
especially the levels on the telescope and the plates ; the cross-axis ot 
the telescope should be exactly horizontal, and the index error of the 

vertical circle care- 
fully determined. 

Second. — The 

polar axis must be 

at right angles to 

the line of collima- 

tion and horizontal axis of main 

telescope. 

To effect this, level the 
instrument carefully and bring 
the bubble of each telescope 
level to the middle of its scale. 
Revolve the solar around its 
polar axis, and if the bubble 
remains central, the adjust- 
ment is complete. If not, 
correct half the movement by 
the adjusting screws at the 
base of the polar axis, and the other half by moving the solar tele- 
scope on its horizontal axis. 

Third. — The line of collimation of the solar telescope and the axis 
of its level must be parallel. 

To effect this, bring both telescopes into the same vertical plane 
and both bubbles to the middle of their scales. Observe a mark 
through the transit telescope, and note whether the solar telescope 
points to a mark above this, equal to the distance between the hori- 
zontal axes of the two telescopes. If it does not bisect this mark, 
move the cross- wires by means of the screws until it does. Gener- 
ally the small level has no adjustments, and the parallelism is effected 
only by moving the cross-hairs. 

The adjustments of the transit and the solar should be fre- 
quently examined, and kept as nearly perfect as possible. 




Fig. 18. — Saegmuller Solar Attachment 



irk 82] INSTRUMENTS — THEIR USE AND ADJUSTMENTS 35 

DIRECTIONS FOR USING THE ATTACHMENT. 

81. First. — Take the declination of the sun as given in the 
" Nautical Almanac " * for the given day, and correct it for refraction 
and hourly change. Incline the transit telescope until this amount 
is indicated by its vertical arc. If the declination of the sun is north, 
depress it ; if south, elevate it. Without disturbing the po.sition of 
the transit telescope, bring the solar telescope into the vertical plane 
of the large telescope and to a horizontal position by means of its 
level. The two telescopes will then form an angle which equals the 
amount of the declination, and the inclination of the solar telescope 
to its polar axis will be equal to the polar distance of the sun. 

Second. — Without disturbing the relative positions of the two 
telescopes, incline them and set the vernier to the co-latitude of 
the place. 

By moving the transit and the solar attachment around their 
respective vertical axes, the image of the sun will be brought into 
the field of the solar telescope ; after accurately bisecting this image 
the transit telescope must he in the meridian, and the compass-needle 
indicates its deviation at that place. 

The vertical axis of the solar attachment will then 
point to the pole, the apparatus being in fact a small 
equatorial. 

Time and azimuth are calculated from an observed 
altitude of the sun by solving the spherical triangle 
formed by the sun, the pole, and the zenith of the 
place. The three sides, SP, PZ, ZS, complements 
respectively of the declination, latitude, and altitude, 
are given; hence we deduce SPZ, the hour-angle, 
from apparent noon, and PZS the azimuth of the sun. 

The solar attachment solves the same spherical triangle by con- 
struction, for the second process brings the vertical axis of the solar 
telescope to the required distance, ZP, from the zenith, while the 
first brings it to the required distance, SP, from the sun. 

OBSERVATION FOR TIME 

82. If the two telescopes, both being in position — one in the 
meridian, and the other pointing to the sun — are now turned on 
their horizontal axes, the vertical remaining undisturbed until each 

*" American Ephemeris and Nautical Almanac" for each year is published 
several years in advance by the United States government. 




36 



PLANE SURVEYING 



[Art. 83 



is level, the angle between their directions (found by sighting on a 
distant object) is SPZ, the time from apparent noon. 

This gives an easy observation for correction of timepiece, reli- 
able within a few seconds. 



TO OBTAIN THE LATITUDE WITH THE SAEGMULLER SOLAR 

ATTACHMENT 

83. Level the transit carefully, point the telescope toward the 
south and elevate or depress the object end, according as the declina- 
tion of the sun is south or north, an amount equal to the declination. 

Bring the solar telescope into the vertical plane of the main 
telescope, level it carefully and clamp it. With the solar telescope 
observe the sun a few minutes before its culmination ; bring its image 
between the two horizontal wires by moving the transit telescope 
in altitude and azimuth, and keep it so by the slow-motion screws 
until the sun ceases to rise. Then take the reading of the vertical 
arc, and correct for refraction due to altitude by the table below. 
Subtract the result from 90°, and the remainder is the latitude sought. 

TABLE I 

Mean Refraction 

Barometer 30 inches, Fahrenheit thermometer 50° 



Altitude 


Eefraction 


Altitude 


Eefraction 


10° 


5M9" 


20° 


2' 39" 


11 


4 51 


25 


2 04 


12 


4 27 


30 


1 41 


13 


4 07 


35 


1 23 


14 


3 49 


40 


1 09 


15 


3 34 


45 


58 


16 


3 20 


50 


49 


17 


3 08 


60 


34 


18 


2 57 


70 


21 


19 


2 48 


80 


10 



The following table, computed by Professor Johnson, C. E., 
Washington University, St. Louis, will be found of considerable 
value in solar compass work. 

" This table is valuable in indicating the errors to which the work 
is liable at different hours of the day and for different latitudes, as 
well as in serving to correct the observed bearings of lines when it 
afterwards appears that a wrong latitude or declination has been 



Art. 84] INSTRUMENTS — THEIR USE AND ADJUSTMENTS 



37 



used. Thus, on the first day's observations I used a latitude in the 
forenoon of 38° 37', but when I came to make the meridian observa- 
tion for latitude I found the instrument gave 38° 39' . This was the 
latitude that should have been used, so I corrected the morning's 
observations for two minutes error in latitude by this table. 

" It is evident that if the instrument is out of adjustment, the 
latitude found by a meridian observation will be in error ; but if 
this observed latitude be used in setting off the co-latitude, the instru- 
mental error is eliminated. Therefore always use for the co-latitude 
that given by the instrument itself in a meridian observation." 



TABLE II 

Errors in Azimuth (by Solar Compass) for One Minute Error in 
Declination or Latitude 



Hour 



11.30 a.m. . . 
12.30 p.m. . . 

11 a.m > 

1 P.M ) 

10 A.M 

2 P.M 

9 A.M I 

3 P.M. .....) 

8 A.M ) 

4 P.M ) 

7 A.M 

5 P.M 

6 A.M ) 

6 P.M. . . * . . ) 



For 1 Min. Error in 
Declination 



Lat. 30° 



Min. 

8.85 
4.46 
2.31 
1.63 
1.34 
1.20 
1.15 



Min. 

10.00 
5.05 
2.61 
1.85 
1.51 
1.35 
1.30 



Lat. 50° 



Min. 

12.90 
6.01 
3.11 

2.20 
1.80 
1.61 
1.56 



For 1 Min. Error in 
Latitude 



Lat. 30° 



Min. 

8.77 
4.33 
2.00 
1.15 
0.67 
0.31 
0.00 



Lat. 40° 



Min. 

9.92 

4.87 
2.26 
1.30 
0.75 
0.35 
0.00 



Lat. 50° 



Min. 

11.80 
5.80 
2.70 
1.56 
0.90 
0.37 
0.00 



Note. — Azimuths observed with erroneous declination or co-latitude may be 
corrected by means of this table by observing that for the line of collimation set 
too high the azimuth of any line from the south point in the direction S. W. N. E. is 
found too small in the forenoon and too large in the afternoon by the tabular amounts 
for each minute of error in the altitude of the line of sight. The reverse is true 
for the line set too low. 

84. From the last three articles we gather that an observation 
with the solar transit involves four quantities ; viz. (1) the hour- 
angle of the sun, or time of day; (2) the declination of the sun: 



38 PLANE SURVEYING [Art. 85 

(3) the latitude of the place of observation ; and (4) the direction of 
the true meridian. The prime object of the solar instrument is to 
find the true meridian, the other three elements being known. 

85. The time of day may be conveniently found by the method 
of Art. 82. The latitude of the place of observation may be deter- 
mined by observations on a circumpolar star (see Art. 147) ; but for 
this particular work it had best be obtained by the method of Art. 83. 

86. To find the Declination of the Sun. — ^The declination of the 
sun is his angular distance north or south of the equator measured on 
an hour-circle (Art. 27, and Fig. 1). About the 20th of March the 
sun crosses the equator, going north, and continues to move north till 
June 21, when he reaches his farthest point north and turns and 
comes south, recrossing the equator about Sept. 20, and continuing 
his southward motion till about Dec. 21, when he turns north again, 
reaching the equator once more on March 20. Thus his declination 
is zero at the time of the vernal and autumnal equinoxes, and it has 
its maximum north (or -f ) value on June 21 and its maximum 
south (or — ) value on Dec. 20. In June and December the sun's 
declination is changing most slowly, while in March and September 
it is changing most rapidly. For solar work, therefore, we need a 
table giving the declination for each hour of the day; these values 
are spoken of as the declination " settings " for the day's work. 

The " American Ephemeris and Nautical Almanac " gives the 
declination of the sun for noon of each day of the year for Green- 
wich and Washington, with the hourly correction for the declination. 
As our " standard " time is so many hours west of Greenwich, it is 
more convenient to use Greenwich declinations. Then the noon 
Greenwich declinations will correspond to declinations at 7, 6, 5, or 
4 o'clock A.M., according as Eastern, Central, Mountain, or Western 
time is used, for these time-belts are respectively 5, 6, 7, and 8 hr. 
west of Greenwich. Now as the standard time seldom differs more 
than 30 min. at most from local time, and as a difference of 30 min. 
could never change the declination more than 30 sec. of arc, it is 
sufficiently accurate to use the standard time of the place of obser- 
vation in applying the hourly change in the declination. Suppose, 
for example, a solar survey is to be made near Nashville, which 
is in the " Central " (90th meridian) time belt. The declination 
given in the "Almanac" is the declination at 6 o'clock a.m. at the 
place of observation. For any other hour, the hourly change in 
declination must be added (algebraically) to this. 



Art. 87] INSTRUMENTS — THEIR USE AND ADJUSTMENTS 



39 



87. To correct the Declination for Refraction. — Declination is 
affected by refraction, which causes rays of light to be bent down- 
ward on entering the earth's atmosphere, thus making the object 
appear higher in the heavens than it really is. The "Nautical 
Almanac " * gives the apparent declination for noon ; for any other 
time of the day, a correction for refraction must be applied. In 
Table III are given the refraction corrections for latitude 40°, and in 
Table IV are given the so-called latitude coefficients, used for getting 
the corrections for any other latitude. To obtain the proper refrac- 
tion correction for a place whose latitude is 36°, for example, we 
multiply the value taken from Table III by the latitude coefficient 
of Table IV corresponding to 36°. 



TABLE III 
Refraction Correction, Lat. 40 c 



J 


INUARY 


Fr 


BRUABY 


March 


April 


May 


Jtxe 




In-. ' " 




hr. ' 






hr. ' " 




hr. ' " 




hr. ' " 




hr. ' "' 


1 


1 1 58 


1 






1 


1 1 03 


1 


3 57 


1 


1 28 


1 


5 1 11 


2 
3 
4 


2 2 16 


2 






2 


2 1 10 


2 


4 1 19 




2 32 


2 




3 3 04 

4 6 23 

1 1 54 


3 
4 


1 1 

2 1 


26 
37 


3 
4 


3 1 27 

4 2 06 

5 4 39 


3 

4 
5 


5 2 18 

1 39 

2 44 


2 
3 


3 39 

4 55 

5 1 30 


3 
4 
5 


1 19 

2 23 

3 30 


5 


2 2 11 


5 






5 


1 59 


6 


3 54 


4 


1 26 


6 


4 43 


6 




6 


3 2 


04 


6 


2 1 06 


7 


4 1 14 


5 


2 30 


7 


5 1 10 


7 


3 2 59 


7 


4 3 


21 


7 


3 1 21 


8 


5 2 08 


6 


3 37 


8 

9 

10 


1 18 

2 22 

3 29 


8 
9 


4 6 01 
1 1 51 


8 
9 


1 1 

2 1 


21 
31 


8 
9 


4 1 56 

5 4 04 


9 
10 


1 36 

2 41 


7 

8 

9 

10 


4 53 

5 1 26 

1 25 

2 29 


10 
11 


2 2 07 


10 
11 


3 1 


56 


10 
11 


1 55 

2 1 02 


11 
12 


3 51 

4 1 10 


11 
12 


4 43 

5 1 09 


12 
13 


3 2 51 

4 5 40 


12 

13 


4 3 
1 1 


04 
16 


12 
13 
14 


3 1 15 

4 1 47 

5 3 34 


13 
14 


5 1 58 
1 34 


11 
12 
13 


3 36 

4 51 

5 1 22 


13 
14 


1 18 

2 22 


14 


1 1 46 


14 


2 1 


25 


15 


2 38 


14 


1 23 


15 


3 29 


15 


2 2 01 


15 


3 1 


48 


15 


1 52 


16 


3 48 


15 


2 27 


16 


4 42 


16 




16 


4 2 


47 


16 


2 58 


17 


4 1 06 


16 


3 34 


17 


5 1 08 


17 


3 2 40 


17 


5 8 


39 


17 


3 1 10 


18 


5 1 49 


17 


4 49 


18 


1 18 


18 


4 5 00 


18 


1 1 


12 


18 


4 1 39 


19 


1 32 


18 


5 1 18 


19 


2 22 


19 


1 1 42 


19 


2 1 


20 


19 


5 3 08 


20 


2 36 


19 


1 22 


20 


3 28 


20 


2 1 56 


20 


3 1 


40 


20 


1 48 


21 


3 45 


20 


2 26 


21 


4 42 


21 


21 


4 2 


31 


21 


2 54 


22 


4 1 02 


21 


3 33 


22 


5 1 08 


22 
23 


3 2 31 

4 4 35 


22 

23 


5 6 
1 1 


49 
07 


22 
23 
24 


3 1 05 

4 1 32 

5 2 51 


23 
24 


5 1 42 
1 30 


22 
23 
24 


4 47 

5 1 15 
1 21 


23 
24 


1 18 

2 22 


24 


1 1 37 


24 


2 1 


15 


25 


2 34 


25 


2 25 


25 


3 29 


25 


2 1 58 


25 


3 1 


33 


25 


1 45 


26 


3 42 


26 


3 32 


26 


4 42 


26 




26 


4 2 


18 


26 


2 50 


27 


4 58 


27 


4 46 


27 


5 1 08 


27 


3 2 22 


27 


5 5 


28 


27 


3 1 01 


28 


5 1 36 


28 


5 1 13 


28 


1 18 


28 
29 


4 4 07 
1 1 32 


28 






28 
29 


4 1 25 

5 2 34 


29 
30 


1 28 

2 32 


29 


1 20 

2 24 


29 


2 22 

3 29 


30 


2 1 44 








30 


1 42 






30 


3 31 


30 


4 43 


3 2 13 








31 


2 47 








4 44 






31 


4 3 41 
















31 


5 1 11 







* "The Solar Ephemeris and Refraction Tables," for the use of surveyors, are 
published yearly by George N. Saegmuller, Washington, D.C., W\ & L. E. Gurley, 
Troy, N.Y., and other instrument-makers. 



40 



PLANE SURVEYING 



[Art. 88 



Table III — Continued 



July 


August 


September 





CTOBER 


November 


December 




hr. ' " 




hr. ' " 




hr. ' ' ' 




hr. ' " 




hr. ' 


»• 




hr. ' " 


1 


5 1 09 


1 




1 


1 39 


1 


1 59 


1 


2 3 


21 


1 


1 1 54 


2 








2 


2 44 


2 


2 1 06 


2 


313 


57 


2 


2 2 11 


3 
4 
5 


1 19 

2 23 

3 30 


2 
3 
4 


1 26 

2 30 

3 37 


3 
4 
5 


3 54 

4 1 14 

5 2 08 


3 
4 
5 


3 1 21 

4 1 56 

5 4 04- 


3 
4 

5 


4 
5 

1 1 


32 


3 

4 


3 2 59 

4 6 01 

5 


6 


4 43 


5 


4 53 


6 


1 42 


6 


1 1 03 


6 


2 1 


44 


5 


1 1 58 


7 


5 1 10 


6 


5 1 26 


7 


2 47 


7 


2 1 10 


7 


3 2 


13 


6 


2 2 16 


8 

9 

10 


1 20 

2 24 

3 31 


7 
8 
9 


1 28 

2 32 

3 39 


8 

9 

10 


3 57 

4 1 19 

5 2 18 


8 

9 

10 


3 1 27 

4 2 06 

5 4 39 


8 
9 

10 


4 3 
5 

1 1 


41 
37 


7 
8 
9 


3 3 04 

4 6 23 

5 


11 


4 44 


10 


4 55 


11 


1 45 


11 


1 1 07 


11 


2 1 


50 


10 


1 2 00 


12 


5 1 11 


11 


5 1 30 


12 


2 50 


12 


2 1 15 


12 


3 2 


22 


11 


2 2 19 


13 
14 
15 


1 21 

2 25 

3 32 


12 
13 
14 


1 30 

2 34 

3 42 


13 
14 
15 


3 1 01 

4 1 25 

5 2 34 


13 
14 
15 


3 1 33 

.4 2 18 
5 5 39 


13 
14 

15 


4 4 
5 

1 1 


07 
42 


12 

13 
14 


3 3 09 

4 6 38 

5 


16 


4 46 


15 


4 58 


16 


1 48 


16 


1 1 12 


16 


2 1 


56 


15 


1 2 01 


17 


5 1 13 


16 


5 1 36 


17 


2 54 


17 


2 1 20 


17 


3 2 


31 


16 


2 2 20 


18 
19 
20 


1 22 

2 26 

3 33 


17 

18 
19 


1 32 

2 36 

3 45 


18 
19 
20 


3 1 05 

4 1 32 

5 2 51 


18 
19 
20 


3 1 40 

4 2 31 

5 6 29 


18 
19 

20 


4 4 
5 

1 1 


35 
46 


17 
18 
19 


3 3 11 

4 6 47 

5 


21 


4 47 


20 


4 1 02 


21 


1 52 


21 


1 1 16 


21 


2 2 


01 


20 


1 2 01 


22 


5 1 15 


21 


5 1 42 


22 


2 58 


22 


2 1 25 


22 


3 2 


40 


21 


2 2 20 


23 
24 
25 


1 23 

2 27 

3 34 


22 
23 
24 


1 34 

2 38 

3 48 


23 
24 

25 


3 1 10 

4 1 39 

5 3 08 


23 
24 
25 


3 1 48 

4 2 47 

5 8 39 


23 

24 

25 


4 4 
5 

1 1 


59 

50 


22 
23 
24 


3 3 11 

4 6 49 

5 


26 


4 49 


25 


4 1 06 


26 


1 55 


26 


1 1 21 


26 


2 2 


06 


25 


1 2 00 


27 


5 1 18 


26 


5 1 49 


27 


2 1 02 


27 


2 1 31 


27 


3 2 


49 


26 


2 2 19 


28 
29 


1 25 

2 29 

3 36 


27 
28 
29 


1 36 

2 41 

3 51 


28 
29 
30 


3 1 15 

4 1 47 

5 3 34 


28 
29 
30 


3 1 56 

4 3 04 
511 01 


28 
29 

30 


4 5 
5 


33 


27 
28 
29 


3 3 09 

4 6 43 

5 


30 


4 51 


30 


4 1 10 








1 1 26 








30 




31 


5 1 22 


31 


5 1 58 






31 


1 37 

2 04 








31 





88. To prepare the Declination Settings for a Day's Work. — From 
the preceding articles, we are now in a position to understand the 
method of preparing the declination settings, which we shall illus- 
trate by two examples. 

(1) Let it be required to prepare a table of declination settings for 
a point whose latitude is 35°, and which lies in the " Central Time " belt, 
for April 4, 1901. 

Since the time is 6 hr. earlier than at Greenwich, the declina- 
tion given in the ephemeris is the declination here at 6 a.m. of the 
same date. This is found to be + 5° 32' 6". Adding the hourly 
change, + 57". 32, this becomes, for 7 a.m., + 5° 33' 3". The latitude 
coefficient is .82 and the refraction correction for 7 a.m. is, therefore, 

.82 x2'18"=l'53". 

Therefore, for 7 a.m., the setting is 5° 34' 56". Notice that to get 
the refraction correction for 8 a.m., we have .82 x 1' 19"= 1'5"„ 



Art. 



INSTRUMENTS — THEIR USE AND ADJUSTMENTS 



41 



Thus we make out the following table, which gives the declina- 
tion settings for the hours during which the work is likely to be 
done : 



Declination 


Settings for Apr. 


4, 1901 


., Lat. 35°, 


Centrai 


, Time 


Hour. 


Declination 


Refr. Cor. 


Settings 


Hour 


Declination 


Refr. Cor. 


Settings 


7 


o / // 

+ 5 33 3 


+ 1 53 


o / // 

5 34 56 


1 


O / II 

+ 5 38 47 


+ 32 


O 1 II 

5 39 19 


8 


+ 5 34 


+ 1 5 


5 35 5 


2 


+ 5 39 44 


+ 36 


5 40 20 


9 


+ 5 34 58 


+ 47 


5 35 45 


3 


+ 5 40 42 


+ 47 


5 4129 


10 


+ 5 35 55 


+ 36 


5 36 31 


4 


+ 5 41 39 


+ 1 5 


5 42 44 


11 


+ 5 36 52 


+ 32 


5 37 24 


5 


+ 5 42 36 


+ 1 53 


5 44 29 



(2) Let it be required to prepare a declination table for a point in 
latitude 45°, in the "Eastern Time" belt, for Oct. 10, 1890. 

The time now is 5 hr. earlier than that of Greenwich, hence the 
declination given in the ephemeris for Greenwich mean noon is 
the declination at our point at 7 A.M. The declination found is 
-6° 43' 56", and the hourly change is - 56'.87. The latitude 
coefficient is 1.20. 

The table then becomes : 



Declination Settings for Oct. 10, 1890, Lat. 45°, Eastern Time 



Hour. 


Declination 


Refr. Cor. 


Settings. 


Hour. 


Declination 


Refr. Cor. 


Settings. 


7 


o / // 

- 6 43 56 


/ // 

+ 5 35 


o / // 

-6 38 21 


1 


o / // 

- 6 49 37 


+ 1 16 


O 1 II 

-6 48 21 


8 


- 6 44 53 


+ 2 31 


-6 42 22 


2 


- 6 50 34 


+ 124 


-6 49 10 


9 


- 6 45 50 


+ 144 


- 6 44 06 


3 


- 6 51 31 


+ 144 


-6 49 47 


10 


- 6 46 47 


+ 124. 


-6 45 23 


4 


- 6 52 28 


+ 2 31 


-6 49 57 


11 


- 6 47 44 


+ 1 16 


-6 46 28 


5 


- 6 53 25 


+ 5 35 


-6 47 50 



If the date be between June 20 and Sept. 20, the declination is 
positive and the hourly change negative, while if it be between Dec. 
20 and March 20, the declination is negative and the hourly change 
positive. The refraction correction is always positive ; that is, it 
always increases numerically the north declinations and diminishes 
numerically the south declinations. The hourly refraction correc- 
tions given in the ephemeris are exact for the middle day of the five- 
day period corresponding to that set of hourly corrections. For the 



42 



PLANE SURVEYING 



[Art. 89 



extreme clays of any such period an interpolation can be made 
between the adjacent hourly corrections, if desired. 



TABLE IV 

Latitude Coefficients 



Lat. 


COEFF. 


LiT. 


COEFF. 


Lat. 


COEFF. 


Lat. 


COEFF. 


15° 


.30 


27° 


.56 


39° 


.96 


51° 


1.47 


16 


.32 


28 


.59 


40 


1.00 


52 


1.53 


17 


.34 


29 


.62 


41 


1.04 


53 


1.58 


18 


.36 


30 


.65 


42 


1.08 


54 


1.64 


19 


.38 


31 


.68 


43 


1.12 


55 


1.70 


20 


.40 


32 


.71 


44 


1.16 


56 


1.76 


21 


.42 


33 


.75 


45 


" 1.20 


57 


1.82 


22 


.44 


34 


.78 


46 


1.24 


58 


1.88 


23 


.46 


35 


.82 


47 


1.29 


59 


1.94 


24 


.48 


36 


.85 


48 


1.33 


60 


2.00 


25 


.50 


37 


.89 


49 


1.38 






26 


.53 


38 


.92 


50 


1.42 







89. To run Lines with the Solar Transit. — Having prepared a table 
of declination settings, set up the instrument over the point, get the 
telescope in the meridian by the method that applies to the instru- 
ment used, then loosen the alidade and direct the line of sight to 
some point in the line ; the reading of the limb will give the required 
bearing (or azimuth) with the true meridian. 

90. Merits and Defects of Solar Instruments. — The day for the use 

of the solar compass has passed. Though it is possible to do more 
accurate work with it than with the needle-compass, it is not likely 
that it will be more accurate, owing to the many precautions that 
must be observed. Instead of the solar compass, the transit, with or 
without a solar attachment, should be used. 

No solar instrument should be used within an hour of sunrise or 
sunset, or within an hour of noon. 

The ordinary solar compass and solar attachments without the 
solar telescope cannot be used in cloudy weather. Herein lies one 
important superiority of Saegmuller's attachment,* for it can be used 
in hazy and cloudy weather, provided the clouds do not entirely 
obscure the sun's disk. Other advantages are its simplicity, the 
ease with which it may be adjusted, and its precision. 

* And other forms having a solar telescope. 



Art. 92] INSTRUMENTS — THEIR USE AND ADJUSTMENTS 



43 



THE Y-LEVEL 

91. There are many kinds of levels, differing more or less in pre- 
cision and cost. The so-called " Dumpy Level " is one of the simpler 
and cheaper varieties. Locke's Hand Level, which is not mounted 
on a tripod, but is held in the hand, is extensively used for rough 
preliminary work in seeking a location for a railroad. 

The Drainage Level, the Architect's Level, etc., are manufactured 
for special use, as their names imply. A transit with vertical arc 




Fig. 19. — Y-Level 

and telescope-bubble attached can be used as a level for any work 
ordinarily required of the surveyor. For more exact work the 
Y-level is the favorite instrument, though this is giving way to 
other forms of precision levels. 

92. The Engineer's Y-Ievel consists essentially of a telescope 
properly supported on a spindle which revolves in a hollow cylinder 
attached to the leveling-head, which in its turn is supported by a 
tripod. Our cut represents a 20-inch Y-level, a sectional view of 
which is given on page 44. The telescope has near its ends two 
rings of bell-metal, turned very truly and of exactly the same 



44 PLANE SURVEYING [Art. 92 a 

diameter. On these rings it revolves in the Y's (so called from their 
shape), or it can be clamped in any position, when the clips of the 
Y's are brought down upon the rings, by pushing in the tapering 
pins. The telescope-bubble is attached to the under side of the 
telescope, its ends being made movable for the purposes of adjust- 
ment. The leveling-head has the same plates and leveling-screws 
described in the account of the transit. 

92 a. Parallax is the apparent motion of the cross-wires about 
the image of the object sighted at, as the eye is moved behind the 
eye-piece, and is due to lack of coincidence of cross- wires and image. 
It is overcome by focussing the eye-piece on the cross-wires so that 
they appear most distinct, and by bringing the image into focus by 
means of the objective. 

To adjust * for parallax when using a level or transit, proceed as 
follows : 

Throw objective out of focus or direct telescope to sky. Move 
eye-piece in and out till the position that gives the most distinct 
vision of the cross-wires is found. Then bring image into focus by 
means of the objective. Test adjustment by shifting the eye behind 
the eye-piece, observing whether there is any apparent movement of 
cross-wires about the image. This adjustment depends upon the 
eye of the observer. Another person, having eyes of a different 
focal range, would have to readjust the eye-piece. The adjustment, 
or focussing, of the objective depends on the distance of the object 
from the instrument, and must be made whenever a sight on an ob- 
ject at a different distance is taken. 

92 b. For a remark on the general subject of adjustments, see 
Art. 55. A Level should always be kept in good adjustment, though 
errors of adjustments may be eliminated by making the plus (or 
back) sights equal to the minus (or fore) sights. 

In ordinary work the leveler can usually tell by the behavior of 
the instrument when it requires adjustment. If the bar or bubble- 
tube is out of adjustment, the bubble will not remain in the center 
as the telescope is revolved. If elevations are taken on two points,, 
first with equal sights and then with very unequal sights, any appre- 
ciable error in the adjustments will become apparent. 

* This is not, properly speaking, an adjustment. 



Art. 93] INSTRUMENTS — THEIR USE AND ADJUSTMENTS 44 a 

ADJUSTMENTS OF THE Y-LEVEL 

93. First. — To make the Line of Sight Parallel to the Axis of 
the Bubble. 

This adjustment may be divided into two parts : 

(job) To adjust the Line of Collimation. — Set the tripod firmly, as 
in all adjustments, remove the Y-pins from the clips so as to allow 
the telescope to turn freely, clamp the instrument to the leveling- 
head, and by the leveling and tangent screws bring either of the 
wires upon the clearly marked edge of some object, distant from one 
hundred to five hundred feet. Then, with the hand, carefully rotate 
the telescope halfway round, so that the position of the same wire 
is compared with the object selected. 

Should it be found above or below, bring it halfway back by the 
capstan head screws at right angles with it, always remembering the 
inverting property of the eye-piece ; bring the wire again upon the 
object and repeat the first operation until it will reverse correctly. 
Proceed in the same manner with the other wire until the adjust- 
ment is complete. If it should be found that both wires are much 
out, it will be well to bring both nearly correct before either is 
entirely adjusted. 

When this is effected, unscrew the covering of the eye-piece cen- 
tering screws and move each pair in succession, with a screw-driver, 
until the wires are brought into the center of the field of view. The 
inverting property of the eye-piece does not effect this operation and 
the screws are to be moved as it appears they should be. 

To test the correctness of the centering, rotate the telescope and 
observe whether it appears to shift the position of an object. Should 
any movement be apparent, the centering is not perfect.* When 
the centering has once been effected, it remains permanent, the cover 
being screwed on again to protect it from derangement. 

(b) To make the Axis of the Bubble Parallel with the Bearings of 
the Y -Mings. — There are two methods of making this adjustment in 
common use. The better method is the so-called " Peg adjustment." 
For this see Art. 69, page 25. The other method, which is not so 
accurate but takes less time than the peg adjustment, is as follows : 
Bring the bubble into the middle with the leveling-screws, and then, 

* In all telescopes the line of collimation depends upon the relation of the cross- 
wires and objective, and therefore the movement of the eye-piece does not affect the 
adjustment of the wires in any respect. 



44 b 



PLANE SURVEYING 



[Art. 94 



without jarring the instrument, take the telescope out of the Y's 
and reverse it end for end. Should the bubble run to either end, 
lower that end, or, what is equivalent, raise the other by turning the 
adjusting-nuts on one end of the level until, by estimation, half 
the correction is made. Again bring the bubble into the middle by 
the leveling-screws, and repeat the whole operation until the rever- 
sion can be made without causing any change in the bubble. It 
would be well to apply the second adjustment (Art. 94) before this 
adjustment is entirely completed. The adjustment just given may 
be called the reversal method. The peg method is the most exact, 
as the bubble is adjusted to an accurately established horizontal line, 
whereas in the reversal method it is assumed that the axis of the Y's 
and telescope are parallel, which may or may not be true. 

94. Second. — To bring the Axis of the Bubble into the Vertical 
Plane through the Axis of the Telescope. 

Clamp the instrument over either pair of leveling-screws, and 
bring the bubble into the middle of its tube. Turn the telescope 
back and forth in the Y's. If the bubble remains in the middle, no 
adjustment is necessary. If not, bring the bubble halfway back by 
means of the lateral adjusting-screws. Test by repetition. If this 
adjustment is much in error, it should be made approximately right 
before making the first adjustment. 




Fig. 20. — Sectional View of Y-Level 



95. Third. — To make the Axis of the Y's Perpendicular to the 

Vertical Axis of the Instrument. 



Art. 97] INSTRUMENTS — THEIR USE AND ADJUSTMENTS 



45 




This is to enable the telescope to be revolved horizontally with- 
out re-leveling. Having placed the telescope over two of the level- 
ing-screws, level the instrument. Revolve 180° horizontally, and 
correct one-half the movement of the bubble by the Y-nuts on either 
end of the bar, and the other half by the leveling-screws. Repeat 
for a check. 

The first adjustment is by far the most important. 

The use of the level will be explained in the chapter 
on Leveling. 

LEVELING-RODS 



96. There are many forms of leveling-rods. It is 
very important that a rod be made of hard, well-seasoned 
wood that will not warp and will not easily wear. The 
graduations should be well defined and accurately placed. 
There are, in general, two kinds of leveling-rods : the 
Self-reading, or Speaking, and the Target Rod. 

A Self -reading, or Speaking, Rod is one so graduated 
that it may be read directly by the observer who is 
handling the instrument. A Target Rod is furnished 
with a sliding target, which is set by the rodman in 
response to signals made by the observer. After the 
target is set so that its zero is in the line of sight of the 
telescope, its reading is recorded by the rodman. If a 
speaking rod is used, the observer records the elevation 
of the line of sight. 

97. Figure 21 represents a target rod, called the New 
York Rod. This rod is made in two parts, sliding upon 
each other, and is graduated to tenths and hundredths 
of a foot. The rod is sometimes made in three or four 
parts, though two are more common. The front sur- 
face, on which the target moves, reads to 6J ft. on 
the two-part rods. When a greater height is required, 
the horizontal line of the target is fixed at the highest 
graduation, and the upper half of the rod carrying the 
target is moved out of the lower, the reading being then 
obtained by a vernier up to an elevation of 12 ft. This 
vernier and a similar vernier on the target itself enables 
the rod to be read to thousandths of a foot. The target 
is provided with a clamp for making it fast to the rod. 




Fig. 21 



46 



PLANE SURVEYING 



[Art. 98 




98. Self-reading Rods are particularly desirable for use in 
stadia work. If the sights are not too long and the observer's 
eyes are good, a self-reading rod may give results that 
are almost, if not quite, as accurate as target readings. 
Figure 22 represents a stadia rod that the author had 
made for the use of his engineering classes. It is 
modelled after a design used by the United States Coast 
and Geodetic Survey,* with two slight changes in the 
graduations that seemed desirable. f The solid portions 
of the symbols are painted red on a white background. 
Here the principle of the triangle is employed to assist 
the eye in subdividing the graduations. Most self- 
reading leveling-rods are better for short than for long 
sights, but stadia rods are frequently read at long dis- 
tances. The engineer can easily make his own rod, to 
be used for stadia work or as a self -reading leveling-rod. 
The one represented in our cut was a board 5 in. wide, 
12 ft. long, 1 in. thick, stiffened by a strip screwed on 



S 



the back. 



THE PLANE-TABLE 



99. The plane-table is extensively used for topo- 
graphical and map drawing. It is a very satisfactory 
and convenient instrument for filling in the details of 
a topographical survey,- based, as is usually the case, on 
a rigid system of triangles established by the transit. 
Stadia Rod It consists essentially of a drawing-board upon a firm 
° a * ° m ' tripod, having upon its upper surface an alidade provided 
with either a sight-vane or telescope attached to a ruler. 
The telescope has a vertical motion like the telescope of a transit, 
but it has no lateral motion .with respect to the ruler. The whole 
alidade may be moved at pleasure on the board. The details of 
construction vary considerably. The board is made to turn freely 
in azimuth. The plane-table represented in our cut has three 
leveling-screws and a tangent movement in azimuth. The board is 
partially cut away to show the details. A square brass plate with 
two bubbles at right angles and a needle-compass is furnished for 
the purpose of leveling the table and determining the magnetic 
bearing of the lines. This table has an adjustable wooden roller 

* See Baker's "Instruments." 

t Here the graduations are so arranged that in any position the cross-wire will 
have some white field behind it. 



Art. 100] INSTRUMENTS — THEIR USE AND ADJUSTMENTS 



47 



at each end by which the paper is brought down snugly to the 
board or upon which a long sheet can be rolled and unrolled at will. 
Sometimes brass clamps are used to fasten the paper to the board. 
The plumbing-arm shown in the figure determines the point on the 
ground corresponding to a given point on the paper. 

The surveyor familiar with the adjustments of the transit, as 
already given, will have no difficulty in adjusting the plane-table. 




Fig. 23.— Plane-Table 



THE USE OF THE PLANE-TABLE 

100. In the area to be surveyed by means of the plane-table, it is 
necessary that at least two points and the distance between them be 
accurately known beforehand, and the points marked on the ground. 
It is preferable that all the other points be visible from each of the 
two given points, if the method of locating them by intersecting 
lines be used. Suppose now, for illustration, that A and B are the 
two known points, the distance AB having been, carefully measured. 
Having snugly secured to the board a sheet of drawing-paper, 
represented in Figs. 24 and 25 by TT', plot on the paper the line 



48 



PLANE SURVEYING 



[Art. 100 



ab = AB, using the scale to be adopted in the drawing. Set up the 

plane-table so that a shall 
be directly over the point 
A and level it. Then 
place the fiducial edge of 
the ruler along the line ab 
with the telescope point- 
ing toward £>, Fig. 24. 
Orient the table (that is, 
revolve it in azimuth) till 
the telescope points to B, 
bisecting the point by aid of 
the tangent screw. Keep- 
ing the board clamped, 
move the alidade around a 
as a pivot (that is, a must 
be kept on the fiducial 
edge) until the telescope 
points to (7, then draw an 




<3$ v 



S&F 



&> 



.--" D 



indefinite line ac 



i . 



(rep- 



resenting (7) will be some- 
where on this line. In the same way, the other points are sighted, 
and the indefinite lines ae\ 
af, ad' drawn. Here A, 
B, C, I) are the corners of 
a quadrilateral, and E and 
F are a tree and a chimney, 
respectively. 

Next move the instru- 
ment to the point B, set it 
up so that b is exactly over 
the point B, Fig. 25, and 
place the alidade so that 
the fiducial edge of the 
ruler is along ba, the tele- 
scope pointing toward a. 
Then orient the plane-table 
till the line of sight bisects 
a pole held on A. Then, 

keeping the board clamped, N j. 

move the ruler around b fig. 25 




Art. 102] INSTRUMENTS — THEIR USE AND ADJUSTMENTS 49 

(just as before it was moved around a) until D is bisected by the 
line of sight. Draw the line bd", or that portion of the line that 
intersects ad\ thus determining the intersection " d" which corre- 
sponds on the drawing to the point D. In turn, F, JE, and C are 
sighted, and the intersections of bf" , be", be" with af, ae 1 ', ac\ 
respectively, are determined. 

As a check on the work, dne or more of the points thus found, 
say (7, may be marked by a stake on the ground and the plane-table 
set up over that point, and in the same manner as before, the line cd 
may be drawn. This line should pass through the intersection of 
ad' and bd" . It is important to test the work in this way. It 
will be observed that in Fig. 24 the point A is covered by a, and in 
Fig. 25 the point B is covered by b. It is best to draw only such 
portions of the radiating lines as are necessary to determine the 
intersections. Notes furnishing full explanations of what objects 
are located by the lines should be kept in a note-book. 

The telescope is usually fitted with stadia wires. If the dis- 
tances are measured by the stadia (see Chapter V), as is commonly 
done, an object may be determined by a single pointing, the dis- 
tances being plotted off to scale on the proper lines. 

Many problems may be solved by the plane-table. For a fuller 
description of the instrument and its use, the reader is referred to 
an article on the plane-table in the report of the United States Coast 
and Geodetic Survey for 1880, Appendix 13. 

THE SEXTANT 

101. The sextant is the most convenient and the most accurate 
hand instrument for measuring angles. It is invaluable to the mari- 
ner for use on board ship, where none of the instruments that we 
have described can be used. It is extensively used on boats in the 
survey of harbors, the location of buoys, etc. It is sometimes used 
by the surveyor in preliminary surveys on land. The theory of the 
sextant presents no difficulties. It is called a " sextant " because an 
arc of 60° is used in measuring angles. 

DRAWING INSTRUMENTS 

102. For the purpose of plotting, the surveyor needs only a few 
simple instruments whose use can be readily learned. The follow- 
ing list will, in most cases, be all-sufficient : 



50 



PLANE SURVEYING 



[Art. 102 



A drawing-board. A T-square, a triangular scale. 

One triangle, 90°, 60°, 30°. A protractor. 

One triangle, 90°, 45°, 45°. A drawing-pen. 

A pair of dividers, with pen and pencil points. 

A hard pencil, erasing-rubber, and some thumb-tacks. 

103. A convenient size for the drawing-board is 23 x 31 in. 
Occasionally a larger board will be desirable. Well-seasoned pine 
wood is the best material. 

The triangles, which are made of hardwood, india rubber, and 
other material, give at once the simplest and most reliable method 




Fig. 26. — Protractor 

of drawing parallel and perpendicular lines, for which purpose they 
are chiefly used. The hypothenuse should be from 10 to 14 in. long. 
The two are used together, or in connection with a T-square. 

The J -square should be 2 ft. long. 

The triangular scale should be graduated decimally,* and should 
be 12 in. long. 

This rule has six edges, giving a choice of six scales, the smallest 
graduations being ^, ^, ^, ¥ V A' .eV of an inch respectively. It 
is laid on the paper, and distances to the chosen scale are marked 
off directly from it. It is far more convenient than the old flat 
diagonal scale, which required distances to be taken off with the 
dividers and transferred to the paper. 

* Architects and machinists prefer a scale of so many inches, halves, quarters, 
eighths, etc., to the foot. 



Art. 105] INSTRUMENTS — THEIR USE AND ADJUSTMENTS 



51 



104. Protractors. — The protractor is an instrument for measur- 
ing angles, and is made of metal or paper, usually in the form of a 
semicircle. The common brass or german silver protractor, 4 to 6 in. 
in diameter and graduated to half-degrees, is of very little value. 
Paper protractors, 12" to 14" in diameter, are much better, and also 
cheaper. Where very accurate work is desired, a higher grade pro- 
tractor, with movable arm and vernier reading to 1 or 2 min., should 
be used. Such a protractor costs from $12 up. A very convenient 
and reliable steel protractor, which sells for 16.50, is represented 
in our cut, Fig. 26. 

105. To use the Semicircular Protractor. — Suppose it is required 
to lay off the bearing of AB = N. 35° E., and the bearing of BO 
= S. 60° 30' E., Fig. 26 a . 

Lay off on a sheet of paper, sufficiently large to contain the 
drawing to the chosen scale, a north and south line JYS, through 





A, the point where the line is to begin. Then place the 
protractor along NS, as in the figure, so that its diameter 
coincides exactly with NS, its centre being at A. Now 
count off 35° from a to P, and at the 35° mark make a 
dot on the paper at P. AP continued to B will be the required 
line, which should be drawn to the proper linear scale. Move the 
protractor to B, place it as before, but this time count the angle 
60° 30' around from the south point. 



EXERCISES 

1. Measure a line on fairly level ground, from 10 to 20 chains 
in length. Remeasure it in opposite direction and note the differ- 
ence in results, if any. 

2. Measure a line of about the same length on very hilly ground. 
Remeasure it as before and note the difference in results. 



52 PLANE SURVEYING 

3. Lay out on a stretch of level ground the length of a standard 
chain, carefully marking the ends by a line cut in a rock or by a 
tack in the top of a wooden stake. Compare the length of any 
chain or tape that is accessible with the standard thus obtained. 

4. If the length of a line measured with a Gunter's chain that 
is .5 of a link too long is found to be 10.56 chains, what is its true 
length ? 

5. If the chain of the last example is used in measuring the 
boundary of a field that is found to contain 40 A., what is the true 
area of the field ? 

6. If a field is found to contain 30 A. when a chain one link too 
short is used, what is the true area ? 

7. On hilly ground measure a line horizontally, for example, the 
line of Example 2 ; then measure the same line, allowing the chain 
to rest on the slope of the ground, and note the difference. 

8. Adjust a needle-compass. 

9. Adjust a Y -level. 

10. Adjust a transit. 

11. Adjust a Saegmuller attachment. 

12. Set a needle-compass over a point 
0, in view of the five points A, B, C, B, 
and E\ get the bearings of OA, OB, 00, OB, 
and OE ; compute from these bearings the 
angles A OB, BOO, COB, BOB, and BOA, 
and add these angles together. Does their 
sum equal 360°? 

13. Set up a transit over the same point 
0, and measure the horizontal angles. Add 

them together and compare their sum with 360°. 

14. Establish three stations forming a triangle. Measure the 
horizontal angles with the transit and see if their sum is equal to 

180°. 

The above problems are merely suggestive. They may be mul- 
tiplied indefinitely. 




CHAPTER II 



CHAIN SURVEYING 

In the absence of an instrument for measuring angles, effective 
work can be done with simply the chain and ranging poles. 

106. To range out a Line. — This can best be done by three per- 
sons, whom we shall call A, B, and C, each with a ranging pole. 
A and B take their positions on the line, not too close to each 
other; C goes forward and puts his pole in line with A and B; 
then A advances beyond C and puts his pole in line with B and C ; 
then B advances and sets his pole beyond A in line with C and 
A ; and so on. 

107. Over a Hill. — Suppose it is required to range out a line 
from P to Q, a hill intervening so that one cannot be seen from the 
other. 

t 

P- 




Fig. 27 



Let two men, A and 
B, each with a ranging 
pole, select two points 
approximately between 
P and Q, one on the 

slope of the hill toward P, the other over the brow toward #, and 
both visible from P and Q. Let A, from his position at a!, motion 
to B till he puts him in line with P, at V ; then B, at the point b', 
puts A in line with Q; again A, being now at a", puts B in line with 
P ; next B, in his turn, puts A in line with Q. This operation is 
continued until the poles are at a and 5, in line with P and Q. 

108. Through a Wood. — In chaining through wood or brush, 
where one end of a course cannot be seen from the other, it is often 

best to measure a second line, 
as near to the first as convenient, 
t v D but avoiding trees. Suppose that 

FlG - 28 the line AB is more or less ob- 

structed by trees, but that a line AD, free from trees and making 

53 



54 



PLANE SURVEYING 



[Art. 108 



a small angle * with AB, is ranged out and measured. When this 
has been done, DB is measured and AD is divided into a convenient 
number of parts. At the points of division, the lines pq, rs, . . . are 
supposed to be drawn parallel to DB. 

The offsets pq, rs, etc., are known by proportion. Measure these 
offsets approximately parallel to DB and plant stakes, which will be 
on the required line AB. For example, if DB = 12.5 ft., AD = 
500 ft., and At = 200 ft., then 

rs:BD = 200 : 500. 

.-. rs =| BD = 5 ft. 

109. To erect a Perpendicular at Any Point in a Line. 

(a) First Method. — From the given point A, in the line OX, 
lay off, with the chain along OX, AQ — AB. At B and O as centres, 

with a radius considerably greater than 
AC, say one chain-length, describe arcs 
cutting each other at E. Then EA will 
be perpendicular to OX. Why ? 

(b) Second Method. — Hold one 

end of the chain at the point A. Let 
an assistant stretch the chain along A O 
and hold the end of the 20th link exactly on the line AO, at B. 
Let another assistant count off 60* links from .A, hand the end of 
this 60th link to the man at A, and 
catching the chain at the end of the 
45th link (from A), stretch it until 
both parts are taut — OE (25 links) 
and AE (15 links) in the diagram. 
Then evidently, as 25 2 = 20 2 + 15 2 , 
OAE is a right angle and AE is per- 
pendicular to OX. Of course we are not restricted to the combi- 
nation 20, 25, 15, as we simply have to choose three numbers such 
that the square of one is equal to the sum of the squares of the 
other two. 

(c) Third Method. — Hold one end of the chain at A, and fix 
the other end at some point, O, then with O as a centre swing the 
chain around till the end strikes the line OA at E ; mark the point 
E, and, still holding the end at Q, swing the chain around till it 



B 



A C 

Fig. 29 




E 



15 



O B 



20 



A 

Fig. 30 



* If the angle is large, the chances for error are greater. 



Art. 112] 



CHAIN SURVEYING 



55 



reaches B, in line with EC; then AB 
will be the required perpendicular, for 
the angle A being inscribed in a semi- 
circle is a right angle. 

This method is convenient if some 
obstacle prevents the extension of the 
line beyond A, especially if the perpen- 
dicular passes through a 
over a stream. 



building or 




Fig, 31 



Fig. 32 



110. To let fall a Perpendicular from a Given Point to a Given Line. 

(#) First Method. — If the point is within a chain's distance 

of the line, with the chain as a radius 
and A as centre describe an arc cut- 
ting OX at B and C. Find the middle 
point E of BC, and then AE will be 
the perpendicular required. 

(V) Second Method. — The prin- 
ciple of the third method of Art. 109 
may be employed. Let the student show how this may be done. 

111. To let fall a Perpendicular to a Line from an Inaccessible Point. 
— Let P be the point, OX the line. At any point A on the given 
line erect a perpendicular AB 
and prolong it below the line, 
making AC= AB. Find the 
point D on the line OX in 
line with PB and the point 
E in line with PC. Draw 
BE and prolong it to meet 
DC in F; then FP will be 
the required perpendicular. 

112. To run a Line through 
a Given Point Parallel to a 
Given Line. — Let P be the 
point and AB the line. From 
P draw PB perpendicular to AB, and at some other point, B, 

of the line erect a perpendicular BE 
and make it equal to PB. PE will 
then be the required line parallel to 
Fig. 34 AB. 




Fig. 33 



56 



PLANE SURVEYING 



[Art. 112 




Fig. 35 



Or, from P (Fig. 35) draw the oblique line PB terminating 

in the line AB, and find its middle 
point M. From some other point 
of the line AB, as (7, draw CM and 
prolong it to E, making ME = CM. 
Draw PE. It will be the required 
line parallel to AB. 

The solution of the problem in the next article furnishes another 
method. 

113. To construct an Angle Equal to a Given Angle. — Let ABC be 

the given angle. Suppose it is required to draw 

from the point E a line making with EX an 

angle = ABC. 

From B as a centre with some convenient 

radius, as a chain-length, describe the arc AC; 

that is, determine the two points A and C. Measure the distance 

AC. Then from E as a centre with the 
same radius describe an arc DC, and from 
I) as a centre, with radius = AC, describe 
an arc intersecting DG- at F. Draw EF ; 
then DEF will be the angle required. 

Or, if the angle is given in degrees, 

and the surveyor has at hand a table of chords, the method explained 

in Art. 166 may be advantageously employed. 




Fig. 36 




Fig. 37 



OBSTACLES TO ALINEMENT AND MEASUREMENT 

114. One method for prolonging a line beyond an obstacle has 
already been given in Art. 108. Our diagram, Fig. 38, shows a 
second method. 

At two points, A and B, on the 
line to be prolonged erect perpen- 
diculars AE, BF, and make them 
exactly equal. Prolong EF, and at 
two points on it beyond the obstacle, 

such as G- and jET, erect perpendiculars CC and HD equal to AE ; 
then CD will be on AB prolonged. If the distance is wanted, 
measure FG-, which is equal in length to BC. 

115. Third Method. — By equilateral triangles. 

Taking AB on the given line as one side, construct an equi- 
lateral triangle ABC. Prolong A C to D a sufficient distance to pass 





Art. 118] CHAIN SURVEYING 57 

the object ; lay off the equilateral triangle FDF, prolong DF till 
DGi = AD\ at G- construct an equilateral triangle GrKR; then KGr 
will be on the prolongation of AB. 

A convenient way or construct- <r y<r )>\- 

ing an equilateral triangle is as 

follows : Lay off on the line the 

distance AB = 33 links ; hold one 

end of the chain at B and also the 

end of the first link at other end 

of the chain at B ; fix the end of 

the 33d link at A, take hold of the 

66th link and pull the chain till 

both sides are taut ; this will give the vertex C of the equilateral 

triangle. If the distance as well as the alinement is wanted, the 

measure of AD gives also the length of ACr. 

Other methods of passing an obstacle will readily suggest them- 
selves to the thoughtful student. 

116. Nearly all of the above problems can be solved more simply 
and with greater accuracy with an instrument for measuring angles, 
such as the transit. The solutions are so similar that it will not be 
considered necessary to repeat these problems in the chapters on 
Compass and Transit Surveying. 

117. Areas. — With the chain alone one can make all measure- 
ments necessary for obtaining the areas of plane figures, such as 
triangles, parallelograms, and regular polygons in general, though 
in most cases it is advisable to use also a compass or transit, if such 
an instrument is available. An examination of the formulae on 
page 95 will show the student what measurements must be made in 
order to get the area of fields of simple, regular shapes. Some 
problems are given in another place. His college campus, ball- 
ground, or lots in the neighborhood will furnish other examples. 

HEIGHTS 

118. To determine the Height of a Church Spire, Tree, or Other Object. 

- — On a clear day measure the shadow of a 10-foot pole and the 
shadow of the object whose height is sought. The proportion 

Length of shadow of the pole : 10 ft. 

= length of shadow of the tree : height of the tree, 
gives the height of the tree in feet. 



58 PLANE SURVEYING [Art. 118 

In the absence of an angle measurer, this method gives an ap- 
proximate result. Or, the height may be found by using a transit. 

Suppose the height of a tower BA, 
Fig. 40, is required. Set up the tran- 
sit at some point E, preferably about 
as far from the foot of the tower as 
the top of the tower is from the ground. 
Measure the angle of elevation D CA ; 
E £ J then AD is found by the formula AD 

FlG ' 40 = CD tan D CA, and AB = AD + DB. 

CD is the line of sight, and DB is exactly equal to the height of 
the instrument, CE bat only when the ground EB is level. 

EXERCISES 

1. Given a point near a building, mark the point on the founda- 
tion of the building exactly opposite * the given point ; that is, de- 
termine the perpendicular from the given point to the line of the 
building. 

2. Determine a point 90 ft. from the front wall of a house, and 
directly opposite the middle point of the wall. 

3. Through the point thus determined range out a line parallel 
to the house. 

4. Lay off the foundation of a house, 50 ft. by 30 ft., the longer 
side being parallel to a straight road and 300 ft. distant from it. 

5. Measure carefully the dimensions of a large house of irregular 
plan, or of a group of houses. In a near-by field, lay out the founda- 
tion of a similar house or group of houses. 

6. Select a line that intersects a house, and prolong it beyond 
the house by at least two methods. 

7. Stake out a triangle ABC, measure the three sides and com- 
pute its area (Table XII). Using the same triangle, drop a per- 
pendicular from C on AB ; measure the perpendicular and the side 
AB, and compute the area of the triangle (Table XII). 

8. A field in the shape of a trapezoid has its parallel sides equal 
to 20 ch. and 18 ch., and the perpendicular distance between them 
11 ch. What is its area ? 

* Beware of the very common but loose use of the word "opposite." The last 
part of the statement of our problem defines the term as here used. 



CHAIN SURVEYING 



59 



9. The length of a rectangular field containing 10 A. is double 
its breadth. Find its dimensions in chains ; also in feet. 

10. A circular field contains 61,600 sq. ft. ; what is its radius ? 
(Let ir = - 2 Y 2 -.) 

11. What is the length of the side of a square field containing 
one acre? 

12. Lay off a lot in the shape of a hexagon, whose side equals 
1 ch. What is its area ? 

13. The sides of a field ABC- 
BEFGHA, Fig. 41, were meas- 
ured ; then the diagonals BD, BG, 
AG, GB, were ranged out and 
measured ; next the perpendicu- 
lars Bl, Ci, Gk, Hm were deter- 
mined. It was found that BEFG 
was a square. Compute the area 
of the entire field, using the follow- 
ing values, distances being given 
in chains : 

AB = 9 FG= 8 

BC = 20 GH=1S 

CD = 11 HA = 19 

DE = 8 BB = 23.50 

EF = 8 AG = 26.23 

14. Using the chain alone, make a survey of a field of irregular 
shape, as in the last example, and compute its area. 




H 




Fig. 41 




Bl = 


7.80 


Gi = 


9.40 


Gk = 


8.00 


Hm = 


13.00 



CHAPTER III 



COMPASS SURVEYING 



I. FIELD OPERATIONS — ORIGINAL SURVEYS AND RE-SURVEYS 

119. We have seen in Chapter I that the bearing of a line as 
obtained by the needle-compass is an uncertain quantity, owing to 
the variation in the declination of the needle. It is not easy to read 
the needle within 5 min., and, even if it were possible to do so, 
it cannot be relied upon to give an angle with a probable error as 
small as 5 min. But, notwithstanding its manifest defects, the 
needle-compass is a most useful instrument because of its simplicity 
and the rapidity with which it can be used. 

In Art. 59 we have seen how the magnetic bearing of a course is 
obtained. At the outset the following problem is important : 

To determine the angle between tivo courses whose bearings have 
been found. 

120. First Case. — When the courses are run from the same 

angular point. 

(a) Given : 

bearing of PB = N. 78° E., 

bearing of PA = N. 25° E., 

to find the angle APB. 

If NS, Fig. 42, is the north and south 
line, it is evident that 

APB = NPB - NPA = 78° - 25° = 53°. 

So in this case we simply subtract the 
bearings. 

bearing of PC = S. 20° E., 
bearing of PB = N. 78° E., 




(6) Given: 
to find angle BPC. 



60 



Art. 121] 
Here 



COMPASS SURVEYING 

BPC = 180° - (NPB + SPC) 
= 180° - 98° = 82°. 



61 



0) Given: bearing of PA = N. 25° E., 
bearing of PD = S. 48° W., 
to find angle APD (less than 180°). 
Produce DP back to d ; then, 

NPd = SPD = 48°, and 
APd = NPd - NPA = 48° - 25° = 23°, and 
APD = 180° - APd = 180° - 23° = 157°. 
(<*) Given : bearing of PA = N. 25° E., 
bearing of PE= N. 30° W., 

to find angle APE. 

APB=2b° + 30° = 55°. 

It is obvious that rules could be framed for all four cases and 
for those that follow ; but it is a better plan for the student to 
solve any such problem by aid of a diagram until he knows the 
principle so well that he can solve the problem mentally with very 
little likelihood of an error. 

121. Second Case. — When the courses are run in the ordinary 
way as we go around the field. 

In going around a field ABCDA, suppose we have 
the bearings of the sides given, to find the interior 
angles. 

(a) Given : AB, N. 15° E., BO, S. 40° E., to find 
angle ABC. 

Since SB A = NAB, ABC = 15° + 40° = 5o°. 

(5) Given : BC, S. 40° E., CD, S. 10° W., to find 
angle BCD. 

Here BCD = 180° - (40° + 10° = 130°). 

O) Given: CD, S. 10° W., DA, N. 50° W., to find angle CDA. 

CD A = 10° + 50° = 60°. 
00 Given : DA, N. 50° W., AB, N. 15° E., to find DAB. 

DAB = 180° - (50° + 15°) = 115°. 




Fig. 43 



62 



PLANE SURVEYING 



[Art. 122 




FIELD WORK 

122. When a field is to be surveyed, we begin at some convenient 
corner and go entirely around it, taking the bearings of each side with 
the compass and measuring them with the chain or tape. It makes 
no difference whether the field is kept on the right or the left in going 
around it. In the example below, which is used for illustration, the 

field is on the right. 

Let ABODE A, Fig. 44, be 
a field to be surveyed, A the 
corner where the work is to be 
begun, NS the direction of the 
magnetic meridian. The sur- 
veyor should be provided with 
a pocket field book, ruled with 
three columns on the left-hand 
page, the right-hand being re- 
served for remarks. In the first 
column is given the number 
of the station (indicated by let- 
ter or numeral); in the second, 
the bearing; and in the third, 
the distance. Some prefer to put in the station column "A to B" 
"B to (7," etc., where we put simply "i," "J?," etc. 

The bearings in parentheses, in smaller type, are the back-sights, 
or reverse-bearings. A back-sight should alw r ays be taken as a check 
on the first reading ; often it also brings into evidence local attrac- 
tion, or some imperfection of the needle. A back-sight should be 
equal numerically to the fore-sight and differ from it only in the two 
letters by which it is designated. For example, the back-sight from 
B to A should have read N. 75° 25' E. It differed from this reading 
by 7 min., hardly enough to make us suspect local attraction. When 
a small difference like this is left uncorrected, the mean of the two 
readings should be adopted. 

The field work may proceed as follows : 

Having set up the compass over the corner A (if this point is 
accessible), level it, and then let the needle down upon its pivot. 
Sight a rod held at the next corner B, or at least somewhere on 
the line AB. After the needle has settled, the reading S. 75° 25' W. 
gives the bearing of A J5, * which is recorded as below. 

* Remember that, if the south end of the compass-box is kept toward the observer, 
the bearing is always given by the north end of the needle. 



Decimation of the Needle 15 E. 

Scale 20 rds. to 1 in. 

Fig. 44 



Art. 123] 



COMPASS SURVEYING 



63 



Then AB is measured, and its length, 50.04 rd., is put down in 
the distance column. 

Next, having raised the needle off its pivot, carry the compass to 
J5, and set it up. First a back-sight is taken on a pole held at A. 
This reverse reading (N. 75° 32' E.) is given in parentheses just 
above the bearing of the next course. Then a sight is taken on (7, 
and the bearing of BO is found to be N. 10° W, which is duly 
recorded, as is the measured length of BO, 53.72 rd. On moving 



Station 


BEAP.ING 


Distance 
(rods) 


Remarks 


A 
B 
C 
D 
E 


(N. 1°20' W.) 

S. 75° 25' W. 

(N. 75° 32' E.) 

K 10° W. 

(S. 10° E.) 

N. 86° 30' E. 

(S. 86° 40' W.) 

S. 80° 30' E. 

(N\ S0° 30' W.) 

S. 1° 10' E. 


50.04 
53.72 

21.84 
35.92 

35.68 


To a stake. 

To point on rock. 

To white oak. 

To a stake. 

To cross in stone. 



the instrument to the next station, (7, the back-sight to B is found to 
be S. 10° E., which coincides exactly with the forward sight; thus 
the bearing and length of every course are found in order. Under 
the head of " Remarks " each corner should be described as accurately 
and concisely as possible. Much varied information often finds a 
place under the head of "Remarks." The surveyor should form the 
habit of jotting down any bits of information that may possibly throw 
some light upon the survey and be useful 
iii making the plot or settling a disputed 
question at some future time. 

The above method of keeping the field 
notes is the most compact and the best 
form to use in most cases, especially when 
the area is to be obtained by the double 
meridian distance method (see Chapter IV). 

123. A very convenient method for 
keeping the notes when the shape of the 
field is not too complicated, especially if 
offsets are to be taken, is to make a rough 
diagram, as in Fig. 45, the various bearings 




64 PLANE SURVEYING [Art. 124 

and lengths being recorded along the lines to which they belong. 
The distances Dv, Dw, etc., and the offsets would usually be written 
on these lines instead of below the diagram. 

Between I) and y the boundary of the field follows the bank of 
the creek. 



DA = 21.05 ch. 


kv = 0.65 ch. 


Dv = 2.48 ch. 


Iw = 1.20 ch. 


Dw = 3.80 ch. 


mx = 1.00 ch. 


Dx = 4.88 ch. 




Dy = 6.20 ch. 





There are other ways of recording the field notes. It matters not 
so much by what method it is done, provided it be done accurately 
and fully enough, so that the surveyor, in referring to his notes 
later, will have no doubt as to their exact meaning. Another method 
will be explained in Art. 127, where a general example is given. 

124. The boundaries of fields are generally marked by fences, 
and it is seldom in settled countries that the compass can be set on 
the line, or that the chain can be carried on the line. In such 
cases it is necessary to take the bearing of a line parallel to the 

line whose bearing is 
sought, and to meas- 
sure the distance by 
taking rectangular 
offsets, as shown in 
Fig. 46. 

If the bearing 
and length of AB, 
BO cannot be secured directly, owing to the presence of the fence, 
measure off at A and B, perpendicular to AB, the distances Ak and 
Bl of the same length (as a rule it is better not to go farther from 
the boundary than is absolutely necessary). Then set up the com- 
pass at k and a range pole at I ; get the bearing of H, which will be 
the bearing of AB, and measure the distance kl = AB. If there is 
an angle at the next corner, be careful to measure the new corner 
from m opposite B, and not from I. Observe that it matters not 
where the compass is set on a straight line to get the bearing. 

125. Beside the regular variation in the declination of the needle 
local attraction is one fruitful source of error. Temporary, though 
serious, deflection has often been caused by an axe or chain held too 
close to the compass, by a bunch of keys in the pocket, or by the steel 




Fig. 46 



Art. 127] 



COMPASS SURVEYING 



65 




Fig. 47 



frames of the surveyor's spectacles. Local attraction may be caused 
by iron ore in the ground, by an electric current in a wire, or by the 
proximity of a wire fence or some iron structure. The existence of 
local attraction is often revealed by the behavior of the needle. 

126. To correct Local Attraction. — If the back-sights and fore- 
sights have agreed for several stations, and then one is found where 
they do not agree, it may be presumed that 
there is local attraction at the last point. 

For example, suppose (Fig. 47) 

bearing of AB = N. 20° W., 
back-sight at B = S. 20° E., 
bearing of BO = N. 30° E., 
back-sight at = S. 28° W. 

Now, suppose the fore-sight at C to be 
N. 60° E., what is the correct bearing of CD ? 

Here, as the back-sight at B was equal to 
the fore-sight at A, we conclude that there is local attraction at (7, 
which draws the north end of the needle east by two degrees, mak- 
ing it occupy the position NS instead of the true position JY'S'. 
Hence the fore-sight of CD is too small by 2°, and the bearing 
should be N. 62° E. If, after moving the instrument to D, the 
back-sight there is found to be S. 62° W., our correction is verified, 
and the local attraction that existed at C is not present at D. If 
the back-sight at D does not coincide with the corrected fore-sight 
at (7, we must test by the next corner in advance of D. 

It sometimes happens that several adjacent corners are affected, 
and it may be impossible to get a consistent series of bearings without 
the aid of a transit or solar instrument. 

127. G-eneral Example. — As illustrating the method of making 
a compass survey and recording the notes, we take an example of 
a farm, the plot of which (Plate II) is represented on page 68. For 
brevity we give only the notes locating the boundary lines, the 
turnpike, and one lane ; but these are enough to suggest to the 
student the field work necessary to locate definitely the division 
lines, houses, etc. Here the field notes were entered from the 
bottom of the page up. It will be observed that in this survey 
the farm was kept on the left. 

There is a station, represented by A, at each corner, the work 
beginning at "J.," marked by a "stake south line R. R., east side of 



66 



PLANE SURVEYING 



[Ait. 127 





110 




to stake, beginning. 




AH 


N. 85° 10' W. 


along R. R. 




102.40 




to stake on R. R. limits. 




72.41 




stream. 


^^^ 


12.50 
AG 


road 




to sawmill. 




N. 18° 52' E. 






100.75 




to stone monument on 




95 


.50 




bank of creek, 12 rods 




90 


2.04 




above sawmill. 




80 


4.20 








70 


6.70 








60 
50 
40 


8.10 
7.82 
7.30 


■ 


offsets. 

Line follows bank of creek. 




35 


6.40 








30 


8 








20 


12.32 








10 


6.21 J 








AF 


N. 44° E. 






28.68 




to stake on bank of creek. 




4 








Turnpike 








AE 

148.70 


Ea^t 








to stone on west side of 






___ -— stream 


turnpike. 


___ 


114 








AD 


S. 86° E. 






67.51 




to stake near walnut stump. 




AC 


South 






145 




to black oak. 




101.20 


Smith- Allen corner. 






AB 


S. 30° 8' W. 






100 


____- — - stream 


to a stake. 


y 


60.24 




A A , stake south line of 
R. R., east side of turn- 




4 

AA 






pike, nearly opposite de- 


Turnpike 


AT ^T,tO CA/ WT 


pot. Distances in this 
survey given in rods. 




JN. 7o oO W . 




Ad' 




on BE. 




"98 




to line BE. 




Ac' 


South 






87 








AV 


N. 80° W. 


A b' at A b, foot of lane. 



LANE 



.Art J 57] 



COMPASS SURVEYING 



67 





Ac 
89.40 

Ab 
92.60 

51.40 

49 
21 

Aa 


S. 2° 50' E. 


on EF. 
to line EF. 




N. 80° W . 


to stake, centre of pike op- 
posite lane. 


Sawmill 






Bearing of road, S. 85° 15' E. 

front gate, 
stream. 

A a in centre of turnpike 
opposite A A. 


Road 




S. 14° W. 


TURNPIKE 



turnpike, nearly opposite depot." The bearing of the course, AB, 
is N. 75° 50' W., and its length is 100 rd. Along AB are noted 
the distance 4 to far side of turnpike (the width being 4 rd.), and 
60.24, marking the point where a stream crosses the line. Notice 
that in this form of notes the distances read from the preceding 
station, and the last distance entered, immediately below the next A, 
gives the length of the course. The bearing of the next course, BO, 
is found to be S. 30° 8' W., the distance along this course to the 
corner of the Smith and Allen farms, 101.20, is noted, and the entire 
length to the corner at the black oak, 145, is duly recorded ; and so 
the work proceeds. 

F is on the bank of Beaver Creek, and between F and G- the 
boundary follows the bank of the creek. At convenient intervals 
along FGr, every 5 or 10 rd., offsets are taken to the creek. These 
offsets enable us to plot the course of the creek, and by means of the 
trapezoids and triangles formed by them the area outside of FGr is 
readily obtained. 

On returning to A, the instrument is set up in the middle of the 
pike opposite "A," and the pike is run out, only two courses being 
required, as there is only the one bend. Here, on the first course, 
ab, the sawmill road is noted on the left, and the farm lane on the 
right, their bearings being taken and recorded. The private lane 
is also run oat. Notice that the compass is not set over stations c 
and d', unless it is desired to take the reverse bearings as a check. 

To expedite the work, the bearing of the north portion of the 
turnpike could have been taken while the compass was set at U A," 
and that of the south end while the compass was at E. 

If some prominent object can be observed from several corners, 



68 



PLANE SURVEY!^ 



f Art. 127 




Art. 129] COMPASS SURVEYING 69 

it is well to take sights on that object as a check on the work. 
For example, if there happened to be a tall tree, T, near the house, 
visible from A, C, and E, we should get the bearing of AT, CT, and 
ET. The check is furnished by the fact that on the plot the line 
ET should pass through the intersection of AT and CT. Such lines 
are called tie lines. If the transit is the instrument used, these tie 
lines, even in large, complicated surveys, often intersect exactly, 
but where the compass is used such exact intersections cannot be 
expected. 

128. Report of a Survey. — A complete description, with plot, of 
any survey of private or public property made by a county surveyor 
is required by law to be furnished to the clerk of the county court, 
registrar of deeds, or some other official who is authorized to record 
the same. The surveyor is required in most states to give as a part 
of the description of the property the amount of declination of the 
needle, and whether east or west (see Art. 135), and also to append 
the names of the chainmen. 

The bearings and distances are known as the "metes and 
bounds" of a survey. A common form of a "Survey Bill" is as 
follows : 

"Description of the property known as the Buckeye Farm, near 

, in county, surveyed on the 5th day of February, 1901, 

and bounded as follows : 

" Beginning at a point, the northeast corner of John Young's land 
and on J. K. Biddle's line, and marked by a cross X cut in top of a 
stone, thence along said Biddle's line S. 44° 9' E., 21.63 ch. to a stone 
similarly marked ; thence S. 7° 20' E., 16.80 ch. to a stake, about 5 ft. 
north of a small white oak; thence S. 29° 10' W., 40.49 ch. to a 
point on solid rock, the corner of Geo. Fox's land and the old patent 
line; thence along the patent line S. 68° 49' W., 9.31 ch. to a locust 
post, with a spike in its top; thence N.5°16'E., 21.15 ch. to a 
walnut root on south bank of a small creek, a corner of John Young's 
land ; thence, with John Young's line, N. 10° 32' E., 50.71 ch. to the 
beginning, containing 106.9 A. more or less. 

-Magnetic bearings given. JOHN MACKY, Surveyor. 

''Declination, 1° 10' W. 

George Dale, 



David Jones, 



Chainmen.' 



129. Surveying Party. — For the survey of a large farm, the sur- 
veying party should be composed of four persons, — the compass-man, 



70 PLANE SURVEYING [Art. 130 

two chainmen, and a rodman to carry the range-pole. The latter 
should be provided with an axe. If there is much undergrowth, 
another axeman may be needed. In small surveys, a compass-man 
with one assistant can do the work, the compass-man and his assistant 
chaining the course after its bearing has been taken. 



RE-SURVEYS 

130. Besides running new lines in making the original survey 
of a farm, or in dividing an estate into certain parts, the surveyor 
is frequently called upon to re-survey a farm, which involves the 
re-running of old lines. This is as important as any work in con- 
nection with the survey of farms, and is generally the most difficult, 
requiring a great deal of patience and good judgment. The diffi- 
culty arises mainly from two causes : first, the unscientific, if not 
careless, way in which old surveyors did their work, including the 
omission of permanent monuments to mark the corners ; second, the 
uncertain behavior of the magnetic needle. 

The defects of the magnetic needle have been referred to, and 
will be further considered in a subsequent section of this chap- 
ter. It is certain, however, that inconsistencies due to the sins 
of the surveyor have often been attributed to "defects" in the 
needle. 

Now, if the surveyor has been "furnished with the original field 
notes (metes and bounds) of a farm that he is asked to re-survey, 
so as to reestablish the old lines and corners, the cases that may 
arise will probably come under one of the following heads : 

First. — Given the field notes, the declination of the needle, and 
one corner (with or without the date of the survey). 

Second. — Given the field notes, the date of the survey, and one 
corner. 

Third. — Given the field notes, the direction of one course, and 
one corner. 

Fourth. — Given the field notes, and one corner. 

In any case, if no corner is certain, the first step is to endeavor, 
by the use of the surveys of adjacent tracts or some other available 
data, to locate at least one corner. In making the re-survey, the 
surveyor would proceed as follows : 



Art. 132] 



COMPASS SURVEYING 



71 



131. First Case. — The present declination of the needle must 
be obtained (Art. 140). The difference between this value and the 
declination as given in the original notes tells how much the needle 
has varied since the former surve}^ was made. Now, if a vernier 
compass is used, set off the variation thus found in the proper direc- 
tion, and starting at the given corner, run the lines in order, using 
the original bearings. If the instrument is not a vernier compass, 
change all the bearings by the required amount, and using the 
changed bearings begin at the given corner and run the lines in 
order. In either case the original lines ought to be exactly repro- 
duced. Another corner is often discovered and identified in this 
way which furnishes a verification of the work. For example, sup- 
pose the following field notes were taken when the declination was 
3° 50' E.: 



Stations 


Bearings 


Distances 


A 
B 

C 
D 


N. 52° E. 
S. 293° E. 
S. 31|° W. 
N. 61° W. 


10.64 
4.09 

7.68 
7.24 



Corner " JB" known, present declination = 2° 45' E. 

Here the needle has changed 1° 5' (3° 50' - 2° 45'); that is, the 
north end of the needle has, in the interval between the surveys, 
swung west 1° 5'. Hence, with the vernier change the N. and S. 
line of the compass-box 1° 5' in the proper direction. (Clockwise 
or counter-clockwise ?) Or else (if vernier compass is not used) 
change the bearings, making them read N. 53° 5' E., S. 28° 40' E., 
S. 32° 50' W., N. 59° 55' W., and, using these new bearings, run out 
the lines in order, beginning at B. 



132. Second Case. — Given the date of survey, 1880, and the 
field notes. 

We must in this case ascertain, as near as possible, the declina- 
tion in 1880. If other surveys made in the same locality during 
the year 1880 record the declination, that value may be assumed, 
and the difference between that and the present declination is used 
exactly as we have done in the first case. If no such record exists, 
then get from published tables, or other available sources, the annual 
variation of the needle. This multiplied by the time that has 
elapsed since 1880 will give an approximate value for the change 



72 PLANE SURVEYING [Art. 13b 

in the declination. If the annual change is taken as 3 min., and 
the re-survey is made in 1902, then 

3 x (1902 - 1880) = 6Q min. = 1° 6', 

is the variation to be used.* 

133. Third Case. — Given, besides the field notes, the direc- 
tion of one course and one corner. 

The known corner may or may not be adjacent to the course 
whose direction is given. The direction of a line may be well 
established by the foundation of an old stone fence, for example, 
or by certain "marked" trees, or by the fact that a portion of it 
coincides with a known line of some other farm. When this is the 
case, set up the compass at some point on the known line and sight 
a pole held at another point on the line. Note the difference be- 
tween the reading of the needle and the bearing as given in the 
notes ; this difference represents the change in the declination, and 
as we have seen, is all that it is necessary to know in order to re-run 
the old lines. 

134. The fourth case, where we have a corner given and noth- 
ing else except the field notes, does not admit of a ready or certain 
solution. Unless the surveyor can, by running trial lines, discover 
some old marks, or find in the description of adjoining tracts of land 
some data concerning either the date of the field notes or the prob- 
able declination of the needle at the time of the survey, the problem 
cannot be solved. In such a case it becomes the duty of the sur- 
veyor to use due diligence in hunting up old records, and examining 
the ground embracing the tract in question for any possible marks. 
He should by no means despise testimony furnished by old residents, 
which, though seemingly trivial, may give a valuable clew. 

It is beyond the scope of this book to consider the principles 
and laws concerning the re-survey of private lands. The reader 
is referred to " Johnson's Surveying," already mentioned, for an ex- 
cellent brief presentation of this matter ; here he will find given 
the substance of many state supreme court decisions on disputed 
points, f 

* Caution. — If the field notes are found in a deed dated 1880, it does not neces- 
sarily follow that the survey was made at that time ; for the field notes, as often 
happens, may have been copied from a former deed. The author has found this to be 
the case more than once. 

t A large list of rules based on supreme court decisions will be found in Hodg- 
man's "Manual of Land Surveying," 



Art. 136] COMPASS SURVEYING 73 

The foregoing shows plainly how important it is for the surveyor 
to mark all corners with durable monuments, such as a large stone 
buried in the ground below the frost line, with the position of the 
corner indicated by a cross or some other mark. 

II. THE DECLINATION OF THE NEEDLE 

135. The magnetism of the earth, while a very powerful force, 
is still in great measure a mystery ; but we do know that the earth 
apparently acts as a great magnet, the poles of which do not coincide 
with the geographic poles, but are situated some distance from them. 
The magnetic meridian does not in most places coincide with the 
true (or astronomical) meridian, but makes an angle with it. This 
angle is called the declination of the needle. Moreover, this angle is 
not the same for any two places on different magnetic meridians, 
and is not constant for any one place. For example, in the northern 
states the change in declination in an east and west direction will 
average about one minute to the mile, so that the value of the decli- 
nation in the east end of a county may be some forty minutes in error in 
the west end of the same county, while in any one place its value may 
be changing at the rate of from two to six minutes a year (sometimes 
less than 2', occasionally more than 6 r , see Table VII, page 78). 
This change in the declination is called the variation * of the needle. 

136. There is one imaginary line in the Western Hemisphere,! 
passing through the magnetic poles, at every point of which the 
needle points truly north. It is called the Line of No Declination, 
or Agonic Line, because for all points on this line the declination is 
zero. At all points east of the line the declination is west, and at all 
points west of it the declination is east, for east of the agonic line 
the needle points west of true north, and west of it the needle points 
east of true north. In America the line of no declination has been 
moving westward since about the year 1800. An examination of the 
Declination Map % (Plate I) at the beginning of this volume shows 
that in 1910 the agonic line passed just east of Savannah, Ga., 
through the western part of South Carolina and North Carolina, 
nearly through Columbia, S.C., a little east of Asheville, N.C., 

* The terms are found used in rather a confusing way, "variation 1 ' often being 
used for "declination. 1 ' 

t There is a similar one in the Eastern Hemisphere. 

i Reduced from a map published in 1911 by the United States Coast and Geodetic 
Survey. 



74 PLANE SURVEYING [Art. 137 

through the eastern part of Tennessee and Kentucky, just missing 
the extreme western corner of Virginia, thence through Ohio, the 
northeast corner of Indiana, and Michigan, passing near Grand 
Rapids in the latter state. 

137. Historic Note. — It seems that Chinese vessels were guided by the 
magnetic needle certainly as early as the third and fourth centuries of our era. 
Its use seems to have spread from China to western- (European) nations rather 
slowly, as the first notice of the magnetic needle as applied to navigation dated 
back no farther than the eleventh or twelfth century. In China the directive 
property of the needle was made use of on land as early as the twelfth century B.C. 
The declination of the needle must have been observed some centuries before 
Columbus crossed the Atlantic, but prior to that time the fact that the needle did 
not point truly north was supposed to be due to some imperfection of the needle 
in use, and even as late as the sixteenth century the same view was commonly 
held. But to Columbus belongs the credit of discovering not only the line of no 
declination, but the declination itself.* 

He crossed the agonic line in September, 1492, a little west of Fayal Island, of 
the Azores. The needle, which had been pointing east of north, was observed to 
point west of north, a fact which caused much alarm on board ship. The discov- 
ery of the gradual change in the declination, which for any one place had previ- 
ously been supposed to be constant, was made by Gellibrand of England in 1635. 

138. We notice three kinds of variation. 

First, Irregular Variation. — Under this head may be classed 
changes due to magnetic storms which occur at no regular intervals 
and are of varying degrees of intensity and extent. In duration 
these magnetic storms (which must not be confounded with electric 
storms) may be confined to a few hours, or they may last a day, 
or even for several days. Erratic changes of the needle, made 
manifest when the compass is moved, are often due to mineral 
deposits or even more local causes. 

Second, Daily Variation. — The daily variation in the declina- 
tion, which amounts to about 8 min., is caused by the swinging of 
the needle through an arc daily, the north end reaching its extreme 
easterly position about 8 A.M., and its extreme westerly position 
about 1.30 p.m. It has its mean or true declination about 10.30 
A.M. and 8 p.m. This variation is a little greater in summer 
than in winter, as an examination of the following table f will 
show. 

* See Maryland Geological Survey, " First Report on Magnetic Work in Maryland," 
by Dr. L. A. Bauer. 

t Values given to nearest minute. Condensed from a table giving the results of 
observations made at the Washington Magnetic Observatory. 



Art. 139] 



COMPASS SURVEYING 



75 



TABLE V 
Daily Variation of the Needle 



Month 


6 

A.M. 


7 


8 


9 


10 


n 


Noon 


l 


2 


3 


4 


5 


6 

P.M. 


January- 








+ r 


+ 2' 


+ 2' 


+ r 


-r 


— 2' 


-3' 


— 2' 


-1' 








April 


+ 2 


+ 3 


+ 3 


+ 3 


+ 1 


_ 9 


-4 


-4 


-4 


-4 


— 2 


-1 





July 


+ 3 


+ 5 


+ 5 


+ 4 


+ 2 


-1 


-3 


-4 


-5 


-4 


-3 


-1 





October 


+ 1 


+ 2 


+ 3 


+ 3 


+ 1 


-1 


-3 


-3 


-3 


_ o 


-1 









This shows that a difference of at least 8 or 10 min. in the 
direction of a line may be made if no account is taken of the 
daily variation. Surveyors in this country usually neglect this 
variation. 

139. Third, Secular Variation. — The. most important change in 
the direction of the needle is the secular variation, so called because 
it has a period of several centuries. We do not know, however, that 
the change is strictly periodic, for we do not possess at any one sta- 
tion records of a complete swing of the needle. It is not the same 
at different places, as the records that we have, embracing more than 
a " half-swing" at some stations, clearly show. From these records 
we find, for example, at London, Paris, and Rome the time interval 
between dates of extreme positions (half-swings) of the needle is 
about 230 to 240 years, while for stations in the eastern states of this 
country it averages about 150 years. At Paris the maximum easterly 
declination of 9° 36' was reached in the year 1580, and the maximum 
westerly declination of 22° 36' in about 1809, a difference of 32° 12' 
in 229 years. At Baltimore the needle pointed about 6° 6' west in 
1670, and in 1802 it pointed the least amount west, 0° 39' ; hence, 
in an interval of 132 years, the change was 5° 27'. These records of 
Paris and Baltimore illustrate the fact that not only is the period 
different at different places, but the amounts of change are not pro- 
portional to the lengths of the periods. Paris and Baltimore are no 
exceptions to the general rule, and this is convincing proof of the 
importance of investigating with great care the declination at any 
place at the time of a survey with a needle-compass. 

On pages 76, 77, and 78 appear Tables VI and VII, condensed 
from similar tables to be found in Special Publication No. 9, on 
Terrestrial Magnetism, of the United States Coast and Geodetic 
Survey (1911). These tables, in connection with the isogonic 



PLANE SURVEYING 



[Art. 139 



TABLE VI 

Secular Change of the Magnetic Declination in the United States 



Place 



Montgomery, Ala. . 
Holbrook, Ariz. 
Little Rock, Ark. . 
San Jose\ Cal. . . 
Pueblo, Colo. . . 
Hartford, Conn. 
Dover, Del. . 
Washington, D.C. . 
Tampa, Fla. . . . 
Macon, Ga. . . . 
Boise, Idaho . 
Bloomington, 111. . 
Indianapolis, Ind. . 
Des Moines, Iowa . 
Emporia, Kan. . 
Lexington, Ky. . . 
Alexandria, La. 
Portland, Me. . . 
Baltimore, Md. . . 
Boston, Mass. . 
Lansing, Mich. . 
Mankato, Minn. 
Jackson, Miss. . . 
Sedalia, Mo. . . . 
Helena, Mont. . . 
Hastings, Neb. . . 
Elko, Nev. . . . 
Hanover, N.H. . . 
Trenton, N.J. . . 
Santa Rosa, N.M. . 
Albany, N.Y. . . 
Newbern, N.C . . 
Jamestown, N.D. . 
Columbus, Ohio 
Enid, Okla. . . . 
Sumpter, Ore. . 
Philadelphia, Pa. . 
Newport, R.I. . . 
Columbia, S.C. . 
Huron, S.D. . . . 
Chattanooga, Tenn. 
San Antonio, Texas 
Salt Lake, Utah 
Rutland, Vt. . . . 
Richmond, Va. . 
Seattle, Wash. . 
Charleston, W.Va. 
Madison, Wis. . 
Douglas, Wyo. . . 



Lat. 



32 22 
34 55 
34 47 

37 18 

38 14 
4145 

39 09 

38 55 
27 58 
32 51 
43 37 

40 31 

39 47 
4136 
38 25 

38 04 

31 21 

43 39 

39 18 
42 20 

42 44 

44 11 

32 20 

38 43 
46 37 

40 37 
40 51 

43 43 
40 14 

34 56 

42 40 

35 07 

46 54 

39 59 

36 24 

44 45 

39 57 
4130 

34 00 
44 21 

35 01 
29 29 

40 46 

43 37 

37 33 

47 40 

38 21 
43 04 
42 44 



Long. 



86 18 

110 10 

92 18 

121 52 
104 38 

72 40 

75 31 
77 02 

82 28 

83 37 
116 12 

88 59 
86 12 

93 36 

96 12 

84 30 

92 25 

70 17 

76 35 

71 01 

84 32 

93 59 
90 11 
93 14 

112 02 

98 24 

115 46 

72 17 

74 48 

104 41 

73 45 

77 03 
98 43 
83 01 

97 55 
118 13 

75 12 
7120 
81 02 

98 14 

85 18 
98 32 

111 54 
72 58 
77 28 

122 18 
81 38 

89 25 

105 22 



Decl'n 
1780 



4 34E. 



13 37 E. 



4 45 W. 
1 52 W. 
01 W. 
6 15 E. 

5 01 E. 



8 20 W. 
1 25 W. 
6 50 W. 



6 47 W. 
3 06 W. 

5 50 W. 
1 17 E. 



2 44 W. 
6 08 W. 

3 44E. 



6 28 W. 

20 E. 

17 19 E. 



Pecl'n 
1800 



5 24E. 

8 15 E. 
14 32 E. 

4 51 W. 
1 33 W. 

28E. 

6 30 E. 

5 44E. 

5 54E. 
4 44E. 



4 22 E. 

8 04E. 
8 44 W. 
56 W. 
7 01 W. 



7 54E. 



6 49 W. 

2 45 W. 

5 28 W. 
1 44 E. 

3 13 E. 



2 08 W. 
6 19 W. 

4 19 E. 

5 07E. 



6 30 W. 

47E. 

18 27 E. 

2 15 E. 



Deot/k 
1830 



5 47E. 

8 51 E. 
15 30 E. 

5 34 W. 
1 52 W. 

19 E. 

6 15 E. 

5 53E. 

6 33E. 
5 04E. 

10 09 E. 

4 31 E. 

8 41 E. 

9 48 W. 

1 05 W. 

7 47 W. 

4 10 E. 

11 20 E. 

8 24E. 

10 03 E. 

11 39 E. 

7 32 W. 

3 06 W. 

5 50 W. 
135E. 

3 22 E. 



2 22 W. 
7 05 W'. 

4 19 E. 

5 16 E. 



7 13 W. 
38 E. 

19 04 E. 
2 15 E. 

8 34E. 



Art. 139] 



COMPASS SURVEYING 



77 



TABLE VI— Continued 
Secular Change of the Magnetic Declination in the United States 



Decl'n 


Decl'n 


Decl'n 


Decl'n 


Decl'n 


Decl'n 


Decl'n 


1840 


1850 


1860 


1870 


1880 


1890 


1900 


/ 

5 38E. 


5 22 E. 


5 00E. 


4 32 E. 


3 54 E. 


3 15 E. 


2 49 E. 




13 33 E. 


13 44 E. 


13 47 E. 


13 40 E. 


13 25 E. 


13 30 E. 


8 59E. 


8 51 E. 


8 34E. 


8 14 E. 


7 38 E. 


7 01 E. 


6 38 E. 


16 22 E. 


16 45 E. 


17 05 E. 


17 20 E. 


17 24 E. 


17 28 E. 


17 50 E. 




13 47 E. 


13 50 E. 


13 46 E. 


13 31 E. 


13 00 E. 


12 53 E. 
10 23 W. 


6 47 W. 


7 31 W. 


8 09 W. 


8 43 W. 


9 24 W. 


9 49 W. 


2 46 W. 


3 23 W. 


4 03 W. 


4 41 W. 


5 20 W. 


5 51 W. 


6 29 W. 


28 W. 


1 02 W. 


1 41 W. 


2 21 W. 


3 00 W. 


3 36 W. 


4 11 W. 


5 30E. 


5 00E. 


4 28E. 


3 53E. 


3 16 E. 


2 48E. 


2 19 E. 


5 26E. 


5 01E. 


4 29E. 


3 53 E. 


3 14 E. 


2 39 E. 


2 08E. 




18 00 E. 


18 30 E. 


18 45 E. 


18 45 E. 


18 39 E. 


18 51 E. 


6 33E. 


6 18 E. 


5 54 E. 


5 26E. 


4 51E. 


4 10 E. 


3 35 E. 


4 44E. 


4 21 E. 


3 50 E. 


3 20 E. 


2 45E. 


2 05E. 


128E. 


10 30 E. 


10 24 E. 


10 09 E. 


9 44E. 


9 06E. 


8 21 E. 


7 52E. 




11 34 E. 


11 28 E. 


11 15 E. 


10 50 E. 


10 14 E. 


9 56 E. 


4 04 E. 


3 39E. 


3 07E. 


2 33 E. 


1 57 E. 


1 17 E. 


42 E. 


8 48 E. 


8 40E. 


8 24E. 


8 00E. 


7 26 E. ' 


6 55E. 


6 35 E. 


11 07 W. 


11 48 W. 


12 28 W. 


12 58 W. 


13 32 W. 


14 00 W. 


14 26 W. 


1 52 W. 


2 26 W. 


3 05 W. 


3 45 W. 


4 24 W. 


5 00 W. 


5 35 W. 


9 04 W. 


9 48 W. 


10 28 W. 


11 01 W. 


11 30 W. 


11 58 W. 


12 33 W. 


3 46E. 


3 20E. 


2 46 E. 


2 04E. 


1 17 E. 


32E. 


01 W. 


11 42 E. 


11 36 E. 


11 20 E. 


10 54 E. 


10 22 E. 


9 32 E. 


8 57 E. 


8 24E. 


8 13E. 


7 57 E. 


7 31 E. 


6 58 E. 


6 25 E. 


6 01 E. 


10 13 E. 


10 04 E. 


9 46 E. 


9 25 E. 


8 46 E. 


8 05 E. 


7 39 E. 


18 53 E. 


19 18 E. 


19 36 E. 


19 45 E. 


19 34 E. 


19 23 E. 


19 31 E. 


12 07 E. 


12 07 E. 


11 59 E. 


11 42 E. 


11 12 E. 


10 35 E. 


10 14 E. 




17 20 E. 


17 36 E. 


17 41 E. 


17 44 E. 


17 38 E. 


17 49 E. 


8 56 W. 


9 46 W. 


10 31 W. 


11 08 W. 


11 38 W. 


12 01 W. 


12 36 W. 


4 04 W. 


4 43 W. 


5 22 W. 


6 01 W. 


6 41 W. 


7 11 W. 


7 46 W. 




12 43 E. 


12 47 E. 


12 43 E. 


12 25 E. 


12 00 E. 


11 54 E. 


6 53 W. 


7 39 W. 


8 25 W. 


9 04 W. 


9 51 W. 


10 12 W. 


10 50 W. 


50E. 


17 E. 


19 W. 


1 00 w. 


1 40 W. 


2 16 W. 


2 52 W. 




14 10 E. 


14 00 E. 


13 42 E. 


13 13 E. 


12 30 E. 


12 07 E. 


2 53E. 


2 24E. 


1 50 E. 


1 14 E. 


37 E. 


02 W. 


42 W. 




11 13 E. 


11 08 E. 


10 56 E. 


10 33 E. 


10 06 E. 


9 46E. 




19 15 E. 


19 40 E. 


19 58 E. 


20 09 E. 


20 11 E. 


20 26 E. 


3 21 W. 


4 04 W. 


4 46 W. 


5 25 W. 


6 03 W. 


6 43 W. 


7 23 W. 


8 22 W. 


9 06 W. 


9 46 W. 


10 19 W. 


10 50 W. 


11 17 W. 


11 52 W. 


3 44 E. 


3 15 W. 


2 41 W. 


2 03 W. 


1 25 W. 


47 W. 


12E. 


13 06 E. 


13 06 E. 


12 57 E. 


12 40 E. 


12 15 E. 


11 35 E. 


11 08 E. 


4 49 E. 


4 24E. 


3 52 E. 


3 16 E. 


2 36 E. 


2 01 E. 


1 30 E. 


9 48 E. 


9 53E. 


9 48E. 


9 37 E. 


9 19 E. 


8 52 E. 


8 43 E. 




16 25 E. 


16 38 E. 


16 43 E. 


16 38 E. 


16 23 E. 


16 28 E. 


8 29 W. 


9 13 W. 


9 59 W. 


10 39 W. 


11 19 W. 


11 42 W. 


12 17 W. 


05 W. 


36 W. 


1 12 W. 


1 51 W. 


2 29 W. 


3 06 W. 


3 40 W. 


20 49 E. 


21 19 E. 


21 45 E. 


22 06 E. 


22 19 E. 


22 32 E. 


22 54 E. 


1 37 E. 


1 05 E. 


30E. 


12 W. 


51 W. 


1 28 W. 


2 06 W. 


8 34E. 


8 16 E. 


7 49 E. 


7 14 E. 


6 25 E. 


5 40 E. 


5 05 E. 




15 51 E. 


15 59 E. 


15 59 E. 


15 49 E. 


15 19 E. 


15 15 E. 



78 



PLANE SURVEYING 



[Art. 139 



TABLE VII 
Secular Change of the Magnetic Declination — Continued 



Place 



Decl'n 


Deol'n 


Annual Change 


1905 


1910 


in 1910 


2 48E. 


o / 

2 45E. 


-0.7 


13 42 E. 


14 05 E. 


+ 4.3 


6 42E. 


6 49 E. 


+ 1.4 


18 10 E. 


18 32 E. 


+ 4.5 


13 04 E. 


13 19 E. 


+ 3.1 


10 43 W. 


11 11 W. 


+ 5.8 


6 48. W. 


7 13 W. 


+ 4.8 


4 29 W. 


4 51 W. 


+ 4.5 


2 HE. 


2 06 E. 


- 0.8 


2 02 E. 


1 52 E. 


-2.0 


19 08 E. 


19 31 E. 


+ 4.5 


3 29E. 


3 25 E. 


-0.8 


1 18 E. 


1 08 E. 


-1.8 


7 53 E. 


7 57E. 


+ 1.0 


9 59 E. 


10 08 E. 


+ 1.8 


30 E. 


19 E. 


-2.0 


6 40E. 


6 50 E. 


+ 1.8 


14 43 W. 


15 13 W. 


+ 6.0 


5 53 W. 


6 15 W. 


+ 4.5 


12 52 W. 


13 21 W. 


+ 6.0 


15 W. 


27 W. 


+ 2.4 


8 54E. 


9 00 E. 


+ 1.2 


6 03 E. 


6 08 E. 


+ 1.0 


741 E. 


7 46 E. 


+ 1.2 


19 45 E. 


20 02 E. 


+ 3.7 


10 19 E. 


10 28 E. 


+ 2.0 


18 04 E. 


18 27 E. 


+ 4.6 


12 46 W. 


13 16 W. 


+ 6.0 


8 07 W. 


8 33 W. 


+ 5.2 


12 10 E. 


12 29 E. 


+ 3.6 


11 05 W. 


11 31 W. 


+ 5.6 


3 08 W. 


3 25 W. 


+ 3.4 


12 15 E. 


12 24 E. 


+ 1.8 


55 W. 


1 10 W. 


+ 2.8 


9 52 E. 


10 06 E. 


+ 2.6 


20 44 E. 


21 07 E. 


+ 4.6 


7 42 W. 


8 07 W. 


+ 5.0 


12 11 W. 


12 40 W. 


+ 6.0 


00 


12 W. 


+ 2.6 


11 18 E. 


11 28 E. 


+ 1.8 


122E. 


1 12 E. 


-2.0 


8 53E. 


9 09 E. 


+ 3.0 


16 42 E. 


17 03 E. 


+ 4.2 


12 27 W. 


12 57 W. 


+ 6.0 


3 55 W 


4 13 W. 


+ 4.0 


23 14 E. 


23 40 E. 


+ 5.0 


2 23 W. 


2 39 W. 


+ 3.4 


4 56 E. 


4 51 E. 


- 1.0 


15 27 E. 


15 43 E. 


+ 3.2 



Montgomery, Ala. . 
Holbrook, Ariz. 

Little Rock, Ark. . 

San Jose, Cal. . . 

Pueblo, Colo. . . 
Hartford, Conn. 
Dover, Del. . 

Washington, D.C. . 

Tampa, Fla. . . . 

Macon, Ga. . . . 

Boise, Idaho . . . 

Bloomington, 111. . 

Indianapolis, Ind. . 

Des Moines, Iowa . 

Emporia, Kan. . . 
Lexington, Ky. . 
Alexandria, La. . 

Portland, Me. . . 

Baltimore, Md. . . 

Boston, Mass. . . 

Lansing, Mich. . . 
Mankato, Minn. 
Jackson, Miss. . 

Sedalia, Mo. . . . 

Helena, Mont. . . 

Hastings, Neb. . . 

Elko, Nev. . . . 

Hanover, N.H. . . 
Trenton, N.J. 

Santa Rosa, N.M. . 

Albany, N.Y. . . 

Newbern, N.C. . . 

Jamestown, N.D. . 
Columbus, Ohio 

Enid, Okla. . . . 
Sumpter, Ore. 

Philadelphia, Pa. . 

Newport, R.I. . . 

Columbia, S.C. . • 

Huron, S.D. . . . 
Chattanooga, Tenn. 
San Antonio, Texas 

Salt Lake, Utah . 

Rutland, Vt. . . . 

Richmond, Va. . . 
Seattle, Wash. . 

Charleston, W.Va. . 

Madison, Wis. . . 

Douglas, Wyo. . . 



Art. 139] 



COMPASS SURVEYING 



79 



chart* given at the beginning of this book may assist the surveyor 
in forming some idea of what the declination is at his station, in the 
event of his having no true meridian line to test his compass by. 
An examination of the tables should cause him to be cautious in 
adopting a value for the annual change in the declination, and at the 
same time he should remember that there are often local peculiarities 
not brought out in the formulae. 

The following example from actual 
practice (Fig. 48) is given as an illustra- 
tion of the change in the position of a 
farm, if no account is taken of the varia- 
tion of the declination. As a re-survey, 
it comes under our 2d case, Art. 132. 

A deed containing the metes and 
bounds, dated 1880, was furnished the 
surveyor in 1902, with the request that 
he re-survey the farm. Before com- 
mencing work, he looked up a deed given 
to the farmer who owned the land prior 
to 1880. This deed, which was dated 
1849, contained the identical field notes, 
showing that no re-survey was made in 
1880. Going no farther back than this, 
he assumed that the notes were taken in 
1849. 

He had reason to suppose that the 
declination in 1849 was 6° E. The declination in 1902 was 2° 45' E. 
Assuming these values to be correct, the variation in the 53 years 
amounted to 3° 15'; that is, in that interval of time the north end 
of the needle had swung west 3° 15'. 

Accordingly he changed all the bearings by that amount and ran 
out the lines, BC,f CD, etc., represented by the full lines in Fig. 48. 
These were doubtless very near to the actual lines run out in 1849. 
Had he used the old bearings, he would have run out the erroneous 
boundary represented by the dotted lines. Notice that the dotted 
arrow, SJ¥, is the direction of the needle in 1902, and SN' its direc- 

* Isogonic lines are lines of equal declination. This chart, for which the author is 
indebted to the United States Coast and Geodetic Survey, is a decided improvement 
over all previous declination maps. It will be observed that an isogonic line is far from 
being a smooth curve. In fact, the more accurate and numerous the observations are, 
the greater the sinuosities of the isogonic lines. 

1 The corner B was the only corner known. 




80 PLANE SURVEYING [Art. 140 

tion in 1849. The original field notes were as follows : Beginning 
at A, east, 40 poles; B, south, 44 poles; (7, S. 20° W., 29 poles; 
Z), south, 90 poles; E, S. 20° W., 42 poles; F, N. 70° W., 38 
poles; (r, N. 10° W., 190 poles; H, east, 56 poles. After being 
changed, the bearing of AB was S. 86° 45' E., that of BO, S. 
3° 15' W., etc. 

140. To determine the Declination. — The declination of a mag- 
netic needle at any place is obtained by comparing the direction of 
that needle with a true (or geographical) meridian line. Hence, 
there should be established in every town or county a true meridian 
line, and every surveyor should be required by law to test his com- 
pass by this line at least once a year, at the same time of day.* 
This is a law in most, if not all, of the states ; but, unfortunately, in 
many instances it is a dead letter. We shall now consider some 
simple methods of establishing a true meridian line. 

III. THE DETERMINATION OF A TRUE MERIDIAN LINE! 

141. To obtain the direction of the true meridian at any place it 
would suffice to get the direction of Polaris, or some other circum- 
polar star (Art. 23), exactly at its upper or lower culmination 
(Art. 24) ; but, as the star is at such times moving very rapidly 
in azimuth, this calculation would require an accurate knowledge of 
the local time. For this reason it is customary to take observations 
when the star is at either eastern or western elongation, and then to 
lay off to the west or east, as the case may be, an angle equal to 
the azimuth of the star (Art. 20). This is done in the first two 
methods given below. 

Polaris, owing to its proximity to the north pole and its being 
so readily distinguished, is the most suitable star for this purpose. 
Its extreme east and west positions are called its eastern and western 
elongations respectively. When it is at elongation, it ceases to 
have any perceptible lateral motion, and moves vertically upward 
at eastern, and downward at western elongation. If the star be 
observed at elongation, the observer's watch may be as much as 

* This remedy for the evils arising from ignorance of the value of the declination 
is said to have been first suggested by David Rittenhouse. 

t The general descriptions of the three methods here given are taken almost 
verbatim from the United States Land Office Manual of Surveying Instructions, 
Washington, 1894. The author has made a few changes and some additional expla- 
nations. 



Art. 141] 



COMPASS SURVEYING 



81 



15 minutes in error without causing any appreciable error in 
the result. 

Table VIII gives the azimuths of Polaris at elongation for any 
year between 1914 and 1924, and for any latitude between + 25° 

and + 50°. 

TABLE VIII 

Azimuths of Polaris when at Elongation for Any Year between 1914 
and 1924, and for any latitude between + 25° and + 50" 



Lat. 


1914 


1915 


1916 


1917 


1918 


1919 


1920 


1921 


1922 


1923 


o 


o 1 


/ 


/ 


/ 


/ 


o / 


o / 


o / 


/ 


/ 


25 


< 16.4 


16.0 


15.7 


15.3 


15.0 


14.7 


14.3 


14.0 


13.6 


13.3 


26 


17.0 


16.6 


16.3 


16.0 


15.6 


15.3 


14.9 


14.7 


14.2 


13.9 


27 


17.7 


17.3 


17.0 


16.6 


16.3 


15.9 


15.6 


15.2 


14.9 


14.6 


28 


18.4 


. 18.0 


17.7 


17.3 


17.0 


16.6 


16.3 


15.9 


15.6 


15.2 


29 


19.1 


18.8 


18.4 


18.1 


17.7 


17.4 


17.0 


16.6 


16.3 


16.0 


30 


19.9 


19.6 


19.2 


18.8 


18.5 


18.1 


17.8 


17.4 


17.0 


16.7 


31 


20.7 


20.4 


20.0 


19.7 


19.3 


18.9 


18.6 


18.2 


17.9 


17.5 


32 


21.6 


21.2 


20.9 


20.5 


20.1 


19.8 


19.4 


19.1 


18.7 


18.3 


33 


22.5 


22.1 


21.8 


21.4 


21.0 


20.7 


20.3 


19.9 


19.6 


19.2 


34 


23.5 


23.1 


22.7 


22.4 


22.0 


21.6 


21.2 


20.9 


20.5 


20.1 


35 


24.5 


24.1 


23.7 


23.3 


23.0 


22.6 


22.2 


21.8 


21.5 


21.1 


36 


25.5 


25.2 


24.8 


24.4 


24.0 


23.6 


23.3 


22.9 


22.5 


22.1 


37 


26.7 


26.3 


25.9 


25.3 


25.1 


24.7 


24.3 


24.0 


23.6 


23.2 


38 


27.8 


27.4 


27.0 


26.6 


26.2 


25.9 


25.5 


25.1 


24.7 


24.3 


39 


29.0 


28.6 


28.2 


27.8 


27.5 


27.1 


26.7 


26.3 


25.8 


25.5 


40 


30.3 


29.9 


29.5 


29.1 


28.7 


28.3 


27.9 


27.5 


27.1 


26.7 


41 


31.7 


31.3 


30.9 


30.4 


30.0 


29.6 


29.1 


28.8 


28.4 


28.0 


42 


33.1 


32.7 


32.3 


31.9 


31.5 


31.0 


30.6 


30.2 


29.8 


29.4 


43 


34.6 


34.2 


33.8 


33.4 


32.9 


32.5 


32.1 


31.8 


31.2 


30.8 


44 


36.2 


35.8 


35.3 


34.9 


34.5 


34.1 


33.6 


33.2 


32.8 


32.4 


45 


37.8 


37.4 


37.0 


36.6 


36.1 


35.7 


35.3 


34.8 


34.4 


34.0 


46 


39.6 


39.2 


38.7 


38.3 


37.8 


37.4 


37.0 


36.5 


36.1 


35.6 


47 


41.5 


41.0 


40.6 


40.1 


39.7 


39.2 


38.8 


38.3 


37.9 


37.4 


48 


43.4 


43.0 


42.5 


42.0 


41.6 


41.1 


40.7 


40.2 


39.8 


39.3 


49 


45.5 


45.0 


44.5 


44.1 


43.6 


43.1 


42.7 


42.2 


41.7 


41.3 


50 


1 47.7 


1 47.2 


1 46.7 


1 46.2 


1 45.7 


1 45.3 


1 44.8 


1 44.3 


1 43.8 


1 43.4 



In Table IX are given the times of the culminations and elon- 
gations of Polaris for the year 1915. The surveyor selects an 
elongation or culmination that does not occur in the daytime. 



82 



PLANE SURVEYING 



[Art. 141 



TABLE IX 

Local Mean (Astronomical*) Time of the Culminations and 
Elongations of Polaris in the Year 1903 

[Computed t for latitude + 40° and longitude 6 hr. west of Greenwich.] 



Date 


East Elonga- 
tion 


Upper Culmi- 
nation 


West Elonga- 
tion 


Lower Culmi- 
nation 


1915 

January 1 


h in 

51.7 


h. m 

6 46.9 
5 51.6 
4 44.5 
3 49.2 
2 54.0 


h m 

12 42.1 

11 46.8 

10 39.7 

9 44.4 

8 49.2 


h m 

18 44.9 
17 49.6 
16 42.5 
15 47.2 
14 52.0 


January 15 
February 1 
February 15 
March 1 . 










23 52.5 
22 45.3 
21 50.1 

20 54.8 


March 15 












19 59.6 


1 58.8 


7 54.0 


13 56.8 


April 1 
April 15 
Mav 1 . 












18 52.7 
17 57.7 
16 54.8 


51.9 


6 47.1 
5 52.0 
4 49.2 


12 49.9 
11 54.8 
10 52.0 


23 52.9 
22 50.0 


May 15 
June 1 












15 59.9 
14 53.3 


21 55.1 

20 48.5 


3 54.2 
2 47.6 


9 57.0 
8 50.4 


June 15 
July 1 
July 15 
August 1 
August 15 












13 58.5 
12 55.9 
12 01.1 
10 54.5 

9 59.8 


19 53.7 
18 51.1 
17 56.3 
16 49.7 
15 55.0 


1 52.8 
50.2 


7 55.6 
6 53.0 
5 58.2 
4 51.7 
3 56.9 


23 51.5 
22 44.9 
21 50.2 


September 1 
September 15 
October 1 . . 
October 15 . 
November 1 . 








8 53.2 

7 58.3 . 
6 55.5 
6 00.6 
4 53.7 


14 48.4 
13 53.5 
12 50.7 
11 55.8 
10 48.9 


20 43.6 
19 48.7 
18 45.9 
17 51.0 
16 44.1 


2 50.3 
1 55.4 
52.7 


23 53.8 
22 46.9 


November 15 








3 58.6 


9 53.8 


15 49.0 


21 51.8 


December 1 . 








2 55.6 


8 50.8 


14 46.0 


20 48.8 


December 15 








2 00.4 


7 55.6 


13 50.8 


19 53.6 



It will be noticed that for the tabular year two eastern elongations 
occur on Jan. 14, two western elongations on July 13, two upper 
culminations on April 14, and two lower culminations on Oct. 14. 
The lower culmination either follows or precedes the upper cul- 
mination by 11 hr. 58.1 min. 

142. Reduction to Other Dates. — (a) For the years up to 1924 
add the following minutes to the quantities given in Table IX : 

* The astronomical day begins 12 hr. after the civil day ; i.e. commences at noon 
on the civil day of the same date. The hours are counted from noon, from to 24. 

t This table was obtained from one given in "Principal Facts of the Earth's Mag- 
netism," published by the United States Coast and Geodetic Survey in 1909. 



Art. 142] 



COMPASS SURVEYING 



83 



1914 


1916 


1917 


1918 


1919 


1920 


1931 


1922 


1923 


- 1.5 


+ 1.6 
-2.3 


-0.7 


+ 0.9 


+ 2.5 


+ 4.0 

+ 0.1 


+ 1.6 


+ 3.1 


+ 4.5 



(6) To refer to any calendar day other than the 1st and 15th 
of each month, subtract 3.92 min. for every day between it and the 
preceding tabular day, or add 3.92 min. for every day between it and 
the succeeding tabular day. 

(c) The correction for longitude is so small that it may usually 
be neglected. It amounts to 0.16 min. subtractive for each hour 
west of 6 hr. 

(cT) To refer to any other than the tabular latitude between the 
limits of 25° and 50° north, add to the time of west elongation 
0.10 min. for every degree south of 40°, and subtract from the time of 
west elongation 0.16 min. for every degree north of 40° ; reverse these 
signs for corrections to times of east elongation. 

(e) To refer the Table to Standard Time. — The time given in 
Table IX is the local time at the place of observation. As hourly 
meridian (or standard) time is now carried at most places in this 
country to the complete exclusion of local time, it will be necessary 
to reduce the tabular local time to standard time. To do this we 
must know the longitude of the place, which can usually be deter- 
mined with sufficient accuracy from a map. If I denotes this longi- 
tude reckoned from Greenwich, we apply to the tabular quantities a 
correction (in minutes) equal to 4(Z — X), where L= 75°, 90°, 105°, 
or 120°, according as the place is in the Eastern, Central, Mountain, 
or Western time belts ; that is, we add a correction of four minutes 
to every degree west of the time meridian. The correction is 
negative if the place is east of its time meridian. 

Note. — The Civil Day, according to the customs of society, commences at 
midnight and comprises 24 hr., from one midnight to the next following. The 
hours are counted from to 12, from midnight to noon, after which they are again 
reckoned from to 12, from noon to midnight. Thus the day is divided into two 
periods of 12 hr. each, the first of which is marked a.m., the last p.m. 

The Astronomical Day commences at noon on the civil day of the same date. 
It also comprises 24 hr. ; but they are reckoned from to 24, from the noon of 
one day to that of the next following. 

The civil day begins 12 hr. before the astronomical day; therefore the first 
period of the civil day answers to the last part of the preceding astronomical day. 

* The upper number before March 1, the lower one after March 1. 



84 PLANE SURVEYING [Art. 142 

Thus, Jan. 9, 2 hr. astronomical time, is also Jan. 9, 2 o'clock p.m., civil time; 
and Jan. 9, 14 hr., astronomical time, is Jan. 10, 2 o'clock a.m., civil time. 

Example. — Suppose the time of the east elongation of Polaris is wanted 
for Oct. 3, 1903, at Sewanee, Tenn. Sewanee is in the Central time belt. Its 
position, as roughly obtained from a map, is assumed to be as follows : latitude, 
35° 13' N. ; longitude, 85° 55' W. From Table IX we find the time of east elon- 
gation for Oct. 1 to be 6 hr. 50.7 min. 

Now, using (c), correction for long. = -f 0.05 min., 

using (d), correction for lat. = — 0.62 min. 

Hence, we have for 





h. 


m. 




Oct. 1, 1903, 


6 


50.7 


tabular time 


Correction for lat. and long 




-0.57 






6 


50.13 


Sewanee local time 


Time correction (see (e)) 




- 16.33 


4 (Z - 90°) 


Oct. 1, 1903 


6 


33.80 


standard time 


Daily change (see (5)), 




-3.94 




Oct. 2, 1903 


6 


29.86 




Oct. 3, 1903 


6 


25.92 





The following methods for determining the position of the north 
pole are easily carried out with the ordinary outfit of a surveyor. 

143. To determine the True Meridian by Observation on Polaris at 
Elongation with the Engineer's or Surveyor's Transit. 

1. Set a stone, or drive a wooden plug firmly into the ground, 
and upon the top thereof make a small distinct mark. 

2. About 30 min. before the time of the eastern or western 
elongation of Polaris, as given by the tables of elongation (Table IX), 
set up the transit firmly, with its vertical axis exactly over the mark, 
and carefully level the instrument. 

3. Illuminate the cross- wires by the light from a bull's-eye lan- 
tern or other source, the rays being directed into the object end of 
the telescope by an assistant. Great care must be taken to see that 
the line of collimation describes a truly vertical plane. 

4. Place the vertical wire upon the star, which, if it has not 
reached its elongation, will move to the right for eastern and to the 
left for western elongation. 

5. While the star moves toward its point of elongation, by means 
of the tangent screw of the vernier plate it will be continually cov- 
ered by the vertical wire until a point is reached where it will appear 
to remain on the wire for some time, then leave it in a direction con- 
trary to its former motion ; this indicates the point of elongation. 



Art. 145] COMPASS SURVEYING 85 

6. At the instant the star appears to thread the vertical wire 
depress the telescope to a horizontal position ; about 300 ft. north 
of the place of observation set a stone or drive a wooden plug, upon 
which, by a strongly illuminated pencil or other slender object 
exactly coincident with the vertical wire, mark a point in the line 
of sight thus determined ; then quickly revolve the vernier plate 
180°, repeat the observation, and as before mark a point in the new 
direction ; then the middle point between the two marks, with the 
point under the instrument, will define on the ground the trace of 
the vertical plane through Polaris at its eastern or western elongation, 
as the case may be. 

7. By daylight lay off to the east or west, as the case may require, 
the proper azimuth taken from Table VIII ; the instrument will then 
define the true meridian,* which may be permanently marked by 
monuments for future reference. 

144. A reflector, such as is shown in our 
diagram, will be found useful for illuminating 
the cross-wires ; a lantern held near and to 
one side of the instrument furnishes the light. 
Such a reflector can be obtained from any 
instrument-maker, or the surveyor may have 
one made of bright tin by his local tinner at a nominal price. 

145. To determine the True Meridian by Observation on Polaris at 
Elongation, with a Plumb-line and Peep-sight. 

1. Attach the plumb-line to a support situated as far above the 
ground as practicable, such as the limb of a tree, a piece of board 
nailed to a telegraph pole, a house, barn, or other building affording 
a clear view in a north and south direction. 

The plumb-bob may consist of some weighty material, such as a 
brick, a piece of iron or stone, weighing four to five pounds, which 
will hold the plumb-line straight and vertical fully as well as one of 
turned and finished metal. 

Strongly illuminate the plumb-line just below its support, by a 
lamp or candle, taking care to obscure the source of light from the 
view of the observer by an opaque screen. 

* To obtain the magnetic declination, take the magnetic bearing of the true merid- 
ian. This bearing is equal in magnitude to the declination • and, if this bearing is 
east, the declination is west ; if the bearing is west, the declination is east. 




86 PLANE SURVEYING [Art. 115 

2. For a peep-sight, cut a slot about one-sixteenth of an inch 
wide in a thin piece of board, or nail two strips of tin, with straight 
edges, to a square block of wood, so arranged that they will stand 
vertical when the block is placed flat on its base upon a smooth 
horizontal rest, which will be placed at a convenient height south 
of the plumb-line, and firmly secured in an east and west direction 
in such a position that, when viewed through the peep-sight, Polaris 
will appear about a foot below the support of the plumb-line. 

The position may be practically determined by trial the night 
preceding that set for the observation. 

3. About 30 min. before the time of elongation, as given 
in the tables of elongation, bring the peep-sight into the same line 
of sight with the plumb-line and Polaris. 

To reach elongation, the star will move off the plumb-line to 
the east for eastern elongation, or to the west for western elongation. 
Therefore, by moving the peep-sight in the proper direction, east or 
west as the case may be, keep the star on the plumb-line until it 
appears to remain stationary, thus indicating that it has reached its 
point of elongation. The peep-sight will now be secured in place 
by a clamp or weight, and all further operations will be deferred 
until the next morning. 

4. By daylight, place a slender rod at a distance of 200 or 300 ft. 
from the peep-sight, and exactly in range with it and the plumb-line ; 
carefully measure this distance. 

Take, from Table VIII, the azimuth of Polaris corresponding 
to the latitude of the station and the year of observation ; find the 
natural tangent of said azimuth and multiply it by the distance 
from the peep-sight to the rod; the product will express the dis- 
tance to be laid off from the rod, exactly at right angles to the 
direction already determined (to the west for eastern elongation, 
or to the east for western elongation), to a point which, with the 
peep-sight, will define the direction of the true meridian with suffi- 
cient accuracy for the needs of local surveyors. 

146. To determine the True Meridian by observing the Transits of 
Polaris and Another Star across the Same Vertical Plane. 

1. A very close approximation to a true meridian may be had by 
remembering that Polaris very nearly reaches the true meridian when 
it is in the same vertical plane with the star Delta (S) in the constel- 
lation Cassiopeia. Using the apparatus just described, plaee the peep- 
sight in line with the plumb-line and Polaris, and move it to the west 



Art. 146] COMPASS SURVEYING 87 

as Polaris moves east, until Polaris and Delta appear upon the plumb- 
line together, and carefully note the time by a clock or watch ; then, 
by moving the peep-sight, preserve its alinement with Polaris and 
the plumb-line (paying no further attention to the other star) ; at the 
expiration of the short interval of time derived from the table below, 
the peep-sight and plumb-line will define the true meridian, which 
may be permanently marked for further use. 

2. This method is practicable only when the star Delta is below 
the pole during the night; when it passes the meridian above the 
pole it is too near the zenith to be of service, and in this case the 
star Zeta (f), the last star but one in the tail of the Great Bear, 
may be used instead. 

Delta (S) Cassiopeia is on the meridian below Polaris and the 
pole at midnight about April 10, and is, therefore, the proper star to 
use at that date, and for some two or three months before and after. 

Six months later the star Zeta (f), in the tail of the Great Bear, 
will supply its place, and should be used in precisely 
the same manner. 

The method given in this article for finding the ^ o % 
true meridian cannot be used with advantage at tf/ * 3 
places below about 38° north latitude, on account of 
the haziness of the atmosphere near the horizon. 

The diagram (Fig. 49), drawn to scale, exhibits Polaris 
the principal stars of the constellations Cassiopeia v ~~ 

and Great Bear, with Delta (8) Cassiopeise, Zeta (f) £ 

of the Great Bear, and Polaris on the meridian, £ 

represented by the straight line ; Polaris being at Cassil 
lower culmination. 

This method is given in Lalande's " Astronomy," 
and was practised by A. Ellicott in 1785 on the Ohio and Pennsyl- 
vania boundary. 

The following table, giving the interval of time before Polaris 
will be exactly on the meridian, is to be used in this method : 

Annual 
Increase 

For Zeta (£) Ursae Majoris in 1912 + 7.1 minutes 0.40 minute 
For Delta (8) Cassiopeia? in 1912 + 8.2 « 0.42 

The foregoing methods for the determination of the true merid- 
ian are excellent in themselves when available, as they answer the 
requirements of the surveyor and give results with all desirable pre- 
cision. They do not require an accurate knowledge of the time, and 






N. Pole 



:>peia 



88 



PLANE SURVEYING 



[Art. 146 



herein lies their principal advantage. The relative motion of the stars 
employed, when near the meridian and the unchangeable azimuth of 
Polaris at elongation (so far as the surveyor is concerned), indicate 
with sufficient exactness the moment when the observation should be 
made. Stormy weather, a hazy atmosphere, or the presence of clouds 
may interfere with or entirely prevent observation when the star is 
either at elongation or on the meridian, and both events sometimes 
occur in broad daylight or at an inconvenient hour of the night. 
There is, however, a simple method applicable at any time (provided 
Polaris is visible) which can often be used by the surveyor when other 
methods fail. For an account of this method the student is referred 
to Johnson's " Theory and Practice of Surveying," or to the " Manual 
of Instructions " (United States Land Office) already mentioned. 

147. Latitude. — Remembering that the latitude of a place is 
equal to the altitude of the pole, the surveyor may determine his 
latitude as follows : 

Simply observe the altitude of a circumpolar star at upper or 
lower culmination, and correct this altitude for the pole distance of 
the star and for refraction. 

Let (f> = latitude, r = refraction, 

d = polar distance, h — altitude ; 

then <j) = h T d — r, 

the minus sign being used for upper and the plus sign for lower 
culminations. 

The refraction r may be obtained from Table I, page 36 ; the 
following table gives d, if Polaris is used : 

TABLE X 

Pole Distance (90° — Declination) of Polaris 



1914 


1915 


1916 


1917 


1918 


1919 


1920 


1° 9'.2 


1° 8'.9 


1° 8'.6 


1° 8'.3 


1° S'.O 


1° 7'.6 


1° 7'.4 



The values given are for the first of January ; for intermediate 
years, interpolate. For other months than January, add to the pole 
distances found for January the following corrections : 

February, OM; March, 0'. 2; April, 0'.3; May, 0'.5; June,0\6; 
July, 0'. 7; August, 0'. 6; September, 0'. 5; October, 0'. 3; November, 
0'.2; December, O'.l. 



CHAPTER IV 
COMPUTATION OF AREAS 









148. The Difference of Latitude (which, for brevity, we shall call 
the "latitude") of a course is the perpendicular distance between 
the east and west lines passing through the beginning and end 
of the course. 

The Difference in Longitude (called the "departure") of a course 
is the perpendicular distance between the meridians passing through 
the extremities of the course. 

149. The "latitude" is called a northern or a southern, according 
as the course runs north or south; the "departure" is an easting or 
a westing, according as the course runs 
east or west. 

A northern is considered positive 
and has the sign -f- ; a southern is 
considered negative and has the sign 
— . Similarly, an easting is plus, and 
a westing is minus. 

The meridian distance of a point 
is its perpendicular distance from any 
assumed meridian. Thus, if NS, 
Fig. 50, is the meridian of reference, 
BE, perpendicular to NS, is the 
meridian distance of B. 

The meridian distance of a line 
is the meridian distance of its middle 
point, and is east or west according 
as the point lies east or west of the assumed meridian. 

In the figure, FC is the meridian distance of the line AB. 

150. If the length and bearing of a course are given, its latitude 
and departure may be found by the following formulae : 

89 




Fig. 50 



90 PLANE SURVEYING [Art. 150 

Let AB be the course; then, AD being parallel to the meridian, 

Latitude * = L. = AD = AB cos DAB, 

Departure = D. = DB = AB sin DAB ; 
that is, 

Latitude = course x cosine of its bearing, 

Departure = course x sine of its bearing. 

For instance, to get the latitude and departure of AB in example, 
page 91, bearings as given on page 63 having been adjusted, we have 
L. = 50.04 x cos 75° 28', 

D. = 50.04 x sin 75° 28'. 

Taking the values of the cosine and sine from the Table of 
Natural Signs, f Table XIX, we have 

L. =50.04x0.25094 = 12.56, 

D. = 50.04 x 0.96800 = 48.44. 

Similarly, for course BO, we have 

L. = 53.72 x cos 10° = 53.72 x 0.98481 = 52.90, 
D. = 53.72 x sin 10° = 53.72 x 0.17365 = 9.33. 

151. The Traverse Table and its Use. — To save the labor of mak- 
ing the calculations of the preceding article for each course, a table 
called a Traverse Table has been prepared from such formulae giving 
the latitude and departure of certain distances for certain angles. 
These tables are usually calculated to every 15 minutes; but as so 
much surveying is now done with the transit, by means of which 
bearings correct to 2' can be obtained, traverse tables are becoming 
antiquated. J The Table of Natural Sines and Cosines (Table XIX) 
will answer the purpose of a traverse table, the sine column giving 
departures for the unit distance, and the cosine column giving the 
latitudes. To adapt the table to this use, the sine and cosine columns 
might be headed "departure for distance unity," and "latitude for 
distance unity," respectively. Using it thus, the result is obtained 
exactly as in Art. 150. No separate traverse table is given in this 
book. 

* Really "difference of latitude," see Art. 148. 

t Or the calculation can be made by logarithms, see Art. 176. 

t A traverse table, computed for every minute of arc and for distances from 1 to 
10, by Major-General J. T. Boileau, F.R.S., is published by D. Van Nostrand Co., 
New York. It is necessarily bulky and expensive. In the opinion of the author very 
little, if any, labor is saved by using a traverse table. 



Art. 152] 



COMPUTATION OF AREAS 



91 



BALANCING THE SURVEY 

152. Error of Closure. — It is evident that in going completely 
around a field back to the starting-point, we have gone just as far 
north as south, and just as far east as west ; therefore, if there are 
no errors, we must have the sum of the northings = sum of the south- 
ings, and the sum of the eastings = sum of the westings. 



X 


Bearings 


5 


Latitudes 


Depar- 
tures 


Cor. 

t 


Balanced 


D.M.D. 


Areas 


N. + 


S.- 


E. + 

21.80 

35.43 

0.78 

58.01 

57.77 

.24 

+ 24^ 
)720 


W.- 

48.44 
9.33 

57.77 

24 

19' 

= li 


03 

1 
1 



1 
1 

.33 
■20 

i 8 


6 

7 
3 
4 
4 

> 
02 


Lat. 


Dep. 


+ 


- 


A 
B 
C 
D 

E 


S. 75°28'W. 
N. 10° W. 
N.86°35'E. 
S. 80°30'E. 
S. 1°15'E. 


50.04 
53.72 
21.84 
35.92 

35.68 


52.90 
1.30 

54.20 
54.16 

.04 

sure = 


12.56 

5.93 

35.67 


-12.57 
+52.89 
+ 1.30 
— 5.94 
-35.68 


-48.50 
- 9.40 

+21.77 
+35.39 
+ 0.74 


+ 67.30 
+ 9.40 
+ 21.77 
+ 78.93 
+115.06 


497.1660 
28.3010 


845.9610 

468.8442 
4105.3408 




Error 


197.20 
of clo 


54.16 

_y/4 s 
1< 


525.4670 5420.1460 
525.4670 

2)4894.6790 

Area _ 2447.3395 
Area ~ sq. rd. 

15 A.1R.7 sq. rd. 



These two relations furnish a means of testing the accuracy of 
the field work. In our example (see tabulated view above) we 
find that the northings exceed the southings by .04 of a rod, and 
the eastings exceed the westings by .24. The meaning of this is that 
there is a gap between the end of the last course and the beginning 
of the first (our diagram, Fig. 44, is not drawn on a scale large 
enough to show this), and that the line filling this gap (completing the 
polygon) has a south latitude = .04, and a west departure = .24, its 
length being = V(.04) 2 +-(.24) 2 = . 2433. The ratio of this length to 
the perimeter of the field is called the error of closure. Hence, in this 

example, error of closure 



or 1 in 802. 



19720' 

This error is not large for compass work. The limit of error 
to be allowed depends upon the importance of the survey. Professor 
Johnson % says " the error of closure for ordinary rolling country 
should not be more than 1 in 300. In city work it should be less 
than 1 in 1000, and should average less than 1 in 5000." 

Unless the land surveyed is of very little value, the surveyor 



* Here distances are given in rods (or poles). t Corrections. 

\ "Theory and Practice of Surveying," J. B. Johnson. 



92 PLANE SURVEYING [Art. 152 

should aim to make the error very much less than 1 in 300 ; and 
yet many county surveyors are so careless in keeping their instru- 
ments in order and insisting on careful horizontal chaining, that 
one often finds the error in the survey of a fertile, valuable farm 
much greater than this limit. Of course, getting the exact area is 
usually a small matter in comparison with locating the bounding 
lines exactly where they belong. 

153. Rules for Balancing the Survey. — Having found the errors in 
latitude and departure, the next step toward getting the area is to 
" balance " the survey, that is, to distribute the error. We give two 
rules for this. 

Rule 1. — The sum of all the courses is to each particular course as 
the whole error in latitude (departure) is to the correction of the cor- 
responding latitude (departure), each correction being so applied as 
to diminish the whole error. 

Rule 2. — The arithmetical sum of all the latitudes (departures) 
is to any one latitude (departure) as the whole error in latitude (de- 
parture) is to the correction of the corresponding latitude (departure)^ 
each correction being so applied as to diminish the whole error. 

The first rule is used when it is assumed that the error is as 
much due to faulty bearings as to erroneous chaining, as is usually 
the case in needle-compass work." The second rule is based on the 
assumption that the error is due almost entirely to errors in chain- 
ing, and this rule should be used if the lines are run with a solar 
attachment or as a traverse with a transit. 

In our example we use Rule 1, and the error in latitude (.04) is 
to be so distributed that a part is to be added to the southings, and 
the remaining part subtracted from the northings. Similarly, for the 
error of .24 in departure, the correction to be applied to the westings 
is additive, that to the eastings is subtractive. In columns 8 and 9 
the corrections are given, and in the 10th and 11th columns the cor- 
rected latitudes and departures are given, only one column being 
devoted to each, and the proper signs given. The corrections must 
be so applied that the sum of the negative latitudes (departures) 
exactly equals the sum of the positive latitudes (departures). 

Theoretically the " corrections " are obtained by the proportions : 

f 19720 : 5004 = 4: x, or x = 1.0 = 1 

For latitude -{ „ „ 

1 19720: 5372= 4: a?, or x = 1.1 = 1 



Art. 156] COMPUTATION OF AREAS 93 



For latitude 



19720 : 2184 = 4: x, or x = 0.5 = 

19720 : 3592 = 4:ar ; or x = 0.7 = 1 

L 19720 : 3568 = 4: x, or x = 0.7 = 1 



Similar proportions for departure give the corrections 6, 7, 3, 4, 
4, taken to the nearest hundredth of a rod. Practically, in an ex- 
ample like this, where the error is not great, the surveyor soon learns 
to put down the corrections mentally after glancing over the lengths 
of the courses. In most cases no two computers will distribute the 
error exactly alike, but the resulting areas will not differ much. 

154. Sometimes it is advisable to attribute a larger part of the 
error to one or more courses than their proportional share. For 
instance, if, in our example, the course AB was hilly and it was 
chained through brush, while the other courses were all on level 
ground, free from undergrowth, it might be legitimate to apply all 
of the error to the latitude and departure of that course, especially 
if in addition to the uncertain chaining there was any reason to 
suppose the bearing of that course doubtful. Sometimes there will 
be a course on which it seems likely that there is twice as much 
chance of error as on some other, while on still other courses there 
is perhaps three times the chance of error. Courses treated in this 
way are said to be " weighted " and to have the weights 2, 3, etc. 
In any case the weight a course should have depends upon the 
judgment of the surveyor. 

155. When the error is excessive, a re-survey is necessary. 
Before making the second survey, go over the balancing carefully 
in order to be sure that the mistake is in the field and not in the 
office. A careful examination of the errors in latitude and departure 
will often be helpful in locating the particular course where the 
larger part of the error probably was made, thus rendering it un- 
necessary to re-survey more than a portion of the boundary. 

DOUBLE MERIDIAN DISTANCES 

156. Having balanced the work, the next step is to calculate the 
double meridian distance (D. M. D.) of each course. While, for this 
purpose, any meridian may be chosen, it will be found more con- 
venient to take as our reference line the meridian through either 
the most westerly or the most easterly station, as the D. M. D.'s will 
then all have the same sign. In the example here given we use the 



94 



PLANE SURVEYING 



[Art. 156 




meridian through the most westerly station. A hurried examination 

of the notes, even when one is not familiar 
with the field, will usually determine the 
most westerly station. 

To deduce rules for computing the 
D. M. D. of any course, we use Fig. 51, 
taking JYS as- our reference meridian. 
Obviously,* we have 

2 JiH = dB. 

2kK = 2ke + 2ef + 2fK= 2 hH+ dB 
+ nA. 

21L = 2ai=2ag + 2gA-2iA=2JcK 
+ nA—pA. 

2 mM= 2ao = 2ai-2pi-2op = 2lL 
-pA-bB. 

Hence, remembering that east departures are +, and west — , we 
have the rules : 

The B. M. B. of the first course is equal to its departure. 

The B. M. B. of the second course is equal to the B. M. B. of the 
first course plus its departure, plus the departure of the second course. 

The D. M. B. of any course is equal to the D. M. B. of the pre- 
ceding course plus the departure of that course, plus the departure of the 
course itself. 

Note. — The D. M. D. of the first course has the same value and the same sign as 
the departure of that course, and the D. M. P. of the last course should have the same 
value and opposite sign of the departure of that course. This serves as a check on the 
work. Thus in the example, of which a tabulated form is given on page 91, C being 
the most westerly course, the D. M. D. of CD ( = + 21.77) is first obtained, and then the 
rest in order by the rules, the D. M.D. of the last (BC) being -f 9.40, which is equal 
to the departure of BC but has the opposite sign. 

157. Area. — If >S r =area ABCD (Fig. 51), it is evident that 
# = the area of the entire figure dBABbd less the sum of the areas 
of the two triangles CdB and CbB, or, numerically, 

S = dBAa + aABb - (CdB + CbB). 
Therefore 

2S=(dB + aA) xBn+(aA + bB) x pB - CdxdB-bCxbB. 

* H, K, Z, and M being the middle points of CD, DA, AB, and BC respectively. 



Art. 159] 



COMPUTATION OF AREAS 



95 



Now dD + aA = D. M. D. of AD, 

aA + bB = D. M. D. of AB, 

dD = D. M. D. of CD, 

bB = D. M. D. of BO, 

and all these values are positive ; Cd and b C are also positive, while 
Dn and pB are negative. 

Hence 
2S=- [(dD + aA) x Dn + ( a A + bB) x pB + Cd x dD + bC x 55]. 

the quantities to be taken with their algebraic signs. 

That is, twice the area of the field is equal to the algebraic sum of the 
products of the double meridian distances of the courses by their corre- 
sponding latitudes, this sum being negative if the field is kept on the 
right in making the survey, and positive, if on the left.* 

In our illustrative example, page 91, the sum of the plus areas 
is found to be 525.4670 sq. rd. and the sum of the minus areas 
5420.1460 ; their difference, 4894.6790, is double the area of the field. 
Hence the area = 2447.3395 sq. rd. = 15 A. 1 R. 7 sq. rd. 

158. As a further illustration, we add another example, for a plot 
of which see Fig. 51. In this example the error of closure (1 in 468) 
is much greater than in the example given on page 91. 



Magnetic 
Bearing 



S.74f°W. 
N.2l|°W. 
N. 64° E. 



DlST. 

(ch.) 



8.86 
19.20 
10.80 
21.05 



59.91 



Latitude 



N.- 



17.87 
4.7-3 



£,rror m l*. == 
Error in closure 



22.60 

22.52 

.08 



S.- 



2.33 



20.19 



22.52 



Departure 



9.71 

5.97 



15.68 
15.58 



W. 



8.55 
7.03 



15.58 



Cor. 



Balanced 



- 2.34 

+ 17.84 

4- 4.72 

- 20.22 



-8.56 
— 7.06 
+ 9.69 

+ 5.93 



D.M.D. 



+ 22.68 
+ 7.06 
+ 9.69 
+ 25.31 



Areas 



125.9504 
45.7368 



171.6872 



53.0712 



511.7682 



564.8394 
171.6872 



,10 = error in D. 



2 ) 393.1522 



V 82 + 102 >O rlin468. 
5991 

Area = 196.576 sq. ch. : 



196.5761 



19 A. 2 R. 25 sq. rd. 



SUPPLYING OMISSIONS 

159. It may happen that on account of the inaccessibility of 
parts of the field, or for other reasons, some of the field notes 

* It is evident, therefore, that the sign of the resulting area has no significance, 
except as it indicates the direction of the field work. 



96 PLANE SURVEYING [Art. 159 

sannot be obtained. In general, two omissions may be supplied. 
We shall consider four cases in which the omitted parts can be 
found by calculation. 

First. — The bearing and length of one course omitted. 

Second. — The bearing of one course and the length of another 
omitted, when the courses are : («) contiguous, (5) separated. 

Third. — The bearings of two courses omitted, when they are : 
(V) contiguous, (5) separated. 

Fourth. — The lengths of two courses omitted, when they are : 
(a) contiguous, (b) separated. 

All four problems may be solved by algebraic equations involving 
trigonometric functions ; but the following trigonometric solutions 
are perhaps simpler. 

160. First Case. — Suppose the bearing and length of CD in 
field on page 95 are omitted. Find the latitudes and departures of 
the other courses. Now, as in a complete survey the sum of the 
northings must be equal to the sum of the southings, and the sum pf 
the eastings to the sum of the westings, if we take the difference 
between the sum of the northings and the sum of the southings, 
in this case = 22.52 — 17.87 = 4.65, a northing, and the difference 
between the sum of the eastings .and the sum of the westings, in this 
case = 15.98 — 5.97 = 9.61, an easting, these numbers (supposing 
there is no error in the field work) give the L. and D. respectively 
of the omitted course ; that is (Fig. 51), 

(7^ = + 4.65, <Z2) = + 9.61, 

and, from the right-angled triangle QdD, we compute the length 
and bearing of CD as follows : 

tan dCD = 5:^ = 2.0666. 



or the bearing of 



4.65 
.-. d6 7 2> = 64°10'30", 

CD = N. 64° 10' E. 
Again, dD = CD sin d CD. 

'•fl»-^-10.6T«L 



Art. 161] 



COMPUTATION OF AREAS 



97 



It will be noticed that this bearing and length of CD do not coincide 
with the values given in the example. This difference is largely 
owing to the fact that all the error made in the field 
work has been concentrated on this one course. For 
this reason it is always best to measure all the courses 
and read all the bearings ; for otherwise there is no 
means of discovering what error has been made in the 
field notes. 

The method of this article gives the means of obtain- 
ing the length and direction of a course which, owing 
to some obstacle, cannot be directly measured. For 
example, suppose that AB is a course in a survey that, 
owing to buildings and trees on the line, cannot be 
directly measured. Run a traverse (the fewer the sides 
the better) AhikB from the beginning A to the end B of AB, 
getting the length and bearing of each side. Then compute, by 
the method just gi^en, the length and bearing of AB. 

161. Second Case. — (a) The defective courses contiguous. — 
In the survey of a field ABCDEF, the incomplete notes of which 
are given below, suppose the length of DE and the bearing of EF 
are omitted. 




Fig. 52 



Stations 


Bearings 


Changed 
Bearings 


DlSTS. 

(ch.) 


Latitudes 


Departures 


N. 


s. 


E. 


W. 


A 


N. 85° W. 


West 


48 









48.00 


B 


S. 76° 30' W. 


S. 71° 30' W. 


5.18 




1.64 




4.91 


C 


N". 8° 30' W. 


N. 13°30'W. 


34 


33.13 






7.65 


D 


N. 5° E. 


North 












E 






345.41 










F 


S. 85° 20' W. 


S. 80° 20' W. 


288.91 




48.51 




284.81 



Suppose the meridian line turned through an angle of 5° clock- 
wise, and change the bearings of all the courses accordingly. The 
object of this is to make the course whose length is missing coincide 
with the meridian line, thus making its departure zero. The shape 
and dimensions of the field are evidently not altered by this suppo- 
sition. Get the L. and D. of all the known courses, using the 
changed bearings. Now, since the D. of BE is zero, the difference 
between the sum of the east D.'s and the sum of the west D.'s of 
the known courses is the D. of the course EF. In this case, as 



98 PLANE SURVEYING [Art. 161 

all the known D.'s are west, their sum, 345.37, is the D. of the course 
EF. We next find the bearing of EF, 

sine (bearing-) = -^±- = , r ' = .99988, 

v 8y distance 345.41 

which gives (see Table XIX) bearing = 89° 1' ; that is, N. 89° V E., 
or S. 85° 53' E. "with reference to the original meridian. As the 
value of the angle is obtained from the. sine, and the sine of an angle 
is the same as the sine of 180° minus the angle, there is an ambiguity 
here, which, however, the surveyor's knowledge of the field will 
usually enable him to remove.* 

Next, to get the L. of EF, we use the formula 

L. = distance x cos (bearing) = 345.41 x .01542 = 5.32, 

and the length of DE< which is its latitude, is at once obtained by 
subtracting the sum of the northings from the sum of the southings, 

thus- D istance 2)^=33.13 + 5.32 -(1.64 + 48.51) 

= 38.45-50.15 = 11.70 ch. 

(5) The defective courses separated. 

It is evident that the method just employed is applicable here. 
We have only to change all the bearings in such a way as to make 
the course the length of which is omitted run north, and proceed 
as in (a). 

162. Third Case. — The bearings of two sides being omitted. 

d (#) The defective courses con- 

C? ""/'' ^^\ tiguous. Find the L.'s and D.'s 

/ -fe£_ _s*jb °f t ne other sides, and then, as in 

/ \ \ the first case, find the length and 

\ \ \ bearing of the line joining the 

\ \ \ extremities of the deficient sides. 

\ \ \ Then, in the triangle thus formed, 

\ h) \f we have the three sides from which 

\ / / to find the angles and thence the 

\ / / bearings. 

\ / / (5) The defective courses 

\ / / separated. Change the places of 

\/ / the sides so as to bring" the de- 

^y, — /q ° 

FlG 53 fective ones next to each other. 

* In this particular case, since the angle is so near 90°, the surveyor, even if he has 
a fair knowledge of the field, may be in doubt whether the course is N. E. or S. E. 



Art. 163] 



COMPUTATION OF AREAS 



99 



Thus (Fig. 53) in the field ABCBEFG, suppose the bearings of BE 
and GA to be missing. Draw Ah parallel and equal to G F, hk 
parallel and equal to FE, then draw kB and kE. kE will evidently 
be parallel and equal to AG. In the figure ABCBkhA, everything 
is known except Bk. Calculate, as in first case, the length and 
bearing of Bk. Then in the triangle BkE, the three sides are 
known, and we can compute the angles, and thence the bearings of 
BE and Ek (or GA), which are required. 

163. Fourth Case. — The lengths of two courses omitted, when 
they are (a) contiguous, (5) separated. 

The method by changing the meridian (Art. 161) may be advan- 
tageously employed here ; or else the method of the last article 
(162). 

EXAMPLES 

Supply the missing parts in the field notes of the surveys given 
below. 

1 2 



Stations 


Bearings 


Distances 
(rd.) 


A 
B 
C 
D 
E 


S. 68° E. 
N. 9° E. 
1ST. 68° W. 

S. 22° W. 


280 

280 
132 

68 



Stations 


Bearings 


Distances 
(rd.) 


A 


S. 6° E. 


60 


B 





55.40 


C 


N. 61° W. 


81.28 


D 





121.72 



Stations 


Bearings 


Distances 
(oh.) 


1 


N. 15° E. 


80 


2 


N. 37° 30' E. 


— 


3 


East 


30 


4 


S. 11° E. 


50 


5 


South 


51 


6 


West 


40 


7 


S. 36° 30 ' W. 


— 


8 


N. 38° 15' W. 


34 



4. Compute the areas of (1), (2), and (3), and draw accurate plot 
of each after supplying the missing parts. 



100 



PLANE SURVEYING 



[Art. 164 



164. Coordinate Method of computing Areas. — This method is use- 
ful in obtaining the areas of large tracts of land made up of smaller 
tracts. It frequently happens that a syndicate or company buys up 
a number of adjacent farms in order to secure the iron ore or coal 
that they contain, or for some other purpose. If every corner is well 
established, this is a convenient way of getting the total area. 

The bearings and distances having been determined, we plot the 
tract and then draw our coordinate axes at right angles to each other 

on the paper. 

It is best, in order 
to avoid change of 
sign, to draw the axes 
so that the entire 
figure shall be con- 
tained in one quad- 
rant. Then draw 
perpendiculars from 
each corner to the 
two lines (axes) re- 
spectively, and, using 
the same scale, meas- 
ure these perpendic- 
ular distances. 

For illustration, 
take an example the 
field notes of which 
are given on page 
106, Ex. 10. We 
first plot the tract 
carefully, Fig. 54, and draw the axes OX and OY,* the former 
running east and west, one chain south of station A, the latter 
running north and south, one chain west of station B. Now it is 
evident that, letting # represent the area, the area of ABCDE is 

S = y a AEy e + y e EBy d + y d BCy c + y c CBy b - y a ABy b . 

,\ 2 S = (x a + x e ) (y e - y a ) + (x e + ar d ) (y d - y e ~) 

+ (x d + 20 (y c - ya) + (x c + x b ) (y b - y^ - (x a + x b } (y b - y a ), 

* These lines are called coordinate axes, and distances measured parallel to the 
jc-axis (the E. and W". line in this case) are called abscissas, and those parallel to the 
#-axis ''the N. and S. line) ordinates. 




Fig. 54 



Art. 166] COMPUTATION OF AREAS 101 

or, rearranging and dividing by 2, 

# = i l x a(y e - yd + Xb(ya - yd + x c (y b - y^ 
+ *d(y c - yd + x e(y d - y„)]> 

Or &~ — \ [y a (Ze - %b) + VbQCa ~ x d + ff e (X b - X d ) 

+ ya(x c - %e) + y e (?d - O] . 

These equations furnish the following rule for finding the area 
from the rectangular coordinates of the corners : 

Multiply the abscissa (ordinate') of each corner by the difference 
between the ordinates (abscisses) of the two adjacent corners, making 
the subtraction in the same direction around the field, and take the half- 
sum of these products. 

The form of reduction, for our example, Fig. 54, is as follows : 



Stations 


Abscissa 

(X) 


Ordinates 

(y) 


Difference between 
Alternate Ordinates 


Double Areas 


A 


3.122 


1.000 


- 2.927 


- 9.1381 


B 


1.000 


5.730 


- 4.674 


- 4.6740 


C 


3.060 


5.674 


+ 1.067 


+ 3.2650 


D 


4.364 


4.663 


+ 2.871 


+ 12.5990 


E 


5.051 


2.803 


+ 3.663 


+ 18.5018 



Plus areas =34.3658 
Minus areas = 13.8121 



2 )20.5537 
Area = 10.2768 sq. ch. 

= 1.028 A. 



PLOTTING 

165. Some description of the drawing instruments necessary for 
making a map or plot of a field has been given (Arts. 102-105). It 
is beyond the scope of this work to enter minutely into the various 
methods of drawing a plot, but a few suggestions and explanations 
may be helpful. For illustration, take the example, Fig. 44, the cor- 
rected notes of which are given in the reduction form on page 91. 

166. Using the bearings and lengths of the courses, we should 
proceed as indicated in Art. 105, if a circular protractor is employed, 
measuring the distances by means of a triangular scale, or other 



102 



PLANE SURVEYING 



[Art. 166 



measure. Corners, or intersections of adjacent lines, are often 
marked by a dot with a little circle around it, as is done in Fig. 44, 
the lines being drawn up to the small circle. The real corner is 
defined in this way better than if the lines were allowed to intersect 
each other.* The direction of the magnetic or true meridian, or 
preferably both, should be given on the plot, and also the scale. 

Instead of using a protractor, the angles may be laid off by means 
of a Table of Chords (Table XV). An example will best explain 
the use of this table. 

Suppose it is required to lay off, at the point J., a line making an 
angle of 20° 10' with AB. From A as a centre, with some conven- 
ient unit radius describe an arc. From the table 
we find that the chord corresponding to 20° 10' is 
0.3502. Hence from B as a centre, with radius 
= .35, describe an arc intercepting the former arc 
at (?, and draw AC; BAG will be the required 
angle. For good results a large scale must be used. This method 
by chords will not give the same degree of precision as that obtained 
by the use of a good vernier protractor. 




A\ 



Fig. 55 



167. Using the deduced latitudes and departures^ the survey may 
readily be plotted by getting the distances of each corner from two 
lines at right angles to each other, the one coinciding with an east 
and west line, the other run- 
ning north and south. For 
simplicity, we shall consider 
our lines of reference drawn 
through (7, the most westerly 
station, using the example 
given on page 91. The work 
is very much simplified by the 
use of cross-section paper as 
in Fig. 5Q, the lines being sup- 
posed to run east and west and 
north and south respectively. 

Rule three columns ; one 
for stations, the second for 
total latitudes, and the third Fig. 56 



-\-—h H M 1 1- 




^g=S = ==::::!:f:::«S=i:=:::::::=:=;:;| 


! — j_j -~== — ..E- 


1 


= ==== : S == = = = === = == = ::=: == == ± ===== = ======= 


===="Jff~===~"===«=="======"====="" 


: :::::S:::::::::::::::::::::~:::::::::::: 


_ ±t _ 


Er = = : =E= = = = = = = = = === = == ==== = ==== = = ====== = == 


:::::::g::::;:::::::::::::::::::::::::::::: 


: :::: J::::::::::::::::::::::::::::::::::: 


tt 


:: :::::SE::::::::::::::::::::::::::::::::::: 


— ±.__±___±± ____ 


== === :=== F :===:::====== == = = == ± = = ==:=:==::= 


:::: ::::$::::::::::::::::::::::::::::::::: : 


_ L 


— — :__ — — if zl___^,^ 


J i_J -;;»" 


t **■"" 


1 _!_„.«=+; 


--'• J ~ 


N^ LJ ' ' 


:::i:::::SEEE:EE::EEE:EE:E:E::::::E::E::=: 



* Because it is difficult to determine the exact intersection of two lines, especially 
if they make a small angle with each other. 

t The values of these after "balancing " had best be taken. 



Art. 169] 



COMPUTATION OF AREAS 



103 



for total departures. Fill the last two columns by beginning at (7, 
and adding (algebraically) the latitudes of the following stations. Do 
the same for the departures, and write down the results as follows : 



Stations 


Total Latitudes from 
(C) 


Total Departures from 

(C) 


c 

D 
E 
A 
B 


0.00 
+ 1.30 

- 4.64 

- 40.32 

- 52.89 


0.00 

+ 21.77 
+ 57.16 
+ 57.90 
+ 9.40 



As a check on the work, it may be noticed that the total latitude 
and total departure of the last course are equal in value to the lati- 
tude and departure of the first course, and have opposite signs. 
Using the values found above, we locate the points B, E, A, B, and 
draw the lines CD, BE, EA, AB, and BC. 

This is a rapid, easy, and accurate method of making the plot. 

168. Copying. — The most satisfactory method of copying is by 
the blue print process. This requires that the final drawing be made 
on tracing cloth (or paper). By exposure to the sunlight, an exact 
copy of the tracing is taken on paper previously prepared. 

This prepared paper and the blue print frames are sold by instru- 
ment makers, whose hand-books or catalogues usually give a full 
explanation of the process. In making the tracing, the scale should 
be drawn, as well as stated ; for, as the paper on which the copy is 
made has to be washed in water after exposure to the sun, it always 
shrinks, and distances can be taken off accurately only by a scale 
drawn on the tracing.* 

In the absence of a blue print outfit, it is possible to copy the 
drawing by means of carbon paper placed under the sheet on which 
the drawing is made, with its carbon side resting on the sheet upon 
which the copy is to be made. 

169. Graphic Method of computing Areas. — Many surveyors com- 
pute the areas of land by plotting the survey, drawing lines dividing 
the polygon into triangles (or other simple figures), and then obtain- 
ing by scale the lengths of one side of each triangle and the perpen- 
dicular dropped upon it from the opposite vertex. 

* It may reasonably be supposed that such a scale has contracted proportionally 
to the rest of the drawing. 



104 PLANE SURVEYING [Art. 169 

The formula, area = | base x the altitude, gives the area of each 
triangle. To obtain good results, the plot should be drawn most 
carefully and to as large a scale as practicable. Owing to lack of 
care in making the drawing, the results are often unreliable, and at 
best they are not as accurate as when the method of calculation 
by double meridian distances is used. 

As an illustration, the field, the plot of which is given as Fig. 44, 
has been divided by the dotted lines into triangles and the altitudes 
of these triangles have been drawn. Using the scale of the drawing, 
we find AB = 55, Bk = 45, M = 23, BB = 55.6, Oh = 20.8. Hence 

Area ABE = | x 55 x 23 = 632.5 
Area ABB = \ x 55 x 45 = 1237.5 
Area BBC = J x 55.6 x 20.8 = 578.84 

Total area = 2448.84 sq. rd. 

= 15 A. 1 R. 8 sq. rd., 
which coincides very nearly with the result obtained on page 91. 



EXAMPLES* 

1. A tract of land is described as follows : Beginning at the 
Valley Pike and running with D. W. Barton N. 50£° W., 220.20 
poles to a Black Oak stump in the lane, then with said Barton 
N. 45 J° E., 173.21 poles to a stone, corner to said Barton in R. L. 
Baker's line, then with Baker S. 44|° E., 139 poles, then S. 48J° E., 
22 poles to the pike, then with it S. 7° W., 44 poles, and S. 30° W., 
127 poles to the beginning, containing two hundred and five acres and 
eighteen square poles. 

Given under my hand this 11th day of August, 1854. 

(Signed) Mahlon Gore, S.F.C. 
Compute the area of the above tract and plot it. 

Compute, by the double meridian distance method, the area of the 
following surveys, and in each case find the error of closure and draw 
the plot. 

* A majority of these examples, as well as those given as exercises under Transit 
Surveying, are actual surveys made by the author's classes. Some are old surveys, 
and the error of closure is often excessive ; but in practice the surveyor will frequently 
meet with similar inexact surveys. 



Art. 169] 



COMPUTATION OF AREAS 



105 



Stations 


Magnetic 
Bearings 


Distances 
(ch.) 


A 


S. 65° W. 


3.48 


B 


S. 58|° W. 


2.20 


C 


S. 69i° W. 


19.12 


D 


N. 22° 15' W. 


4.29 


E 


N. 67° 30' E. 


6.83 


F 


N. 67° 20' E. 


18.04 


G 


S. 22° 15' E. 


4.43 



Stations 


Bearings 


Distances 
(ch.) 


A 


S. 29° 45' W. 


11.64 


B 


N. 55° 58' W. 


3.80 


C 


X. 32° 45' E. 


3.86 


D 


N. 26° 22' E. 


4.00 


E 


N. 56° 10' W. 


13.08 


F 


N. 70° 5' E. 


3.97 


G 


S. 58° 25' E. 


14.29 



Stations 


Bearings 


Distances 
(ch.) 


1 
2 
3 
4 
5 


N. 16° 30' E. 
N. 82° E. 
S. 17° E. 
S. 37° W. 
N. 49° W. 


22 

19.60 
24 
22 

25.20 



Sta. 


Magnetic Bearings 


Distances (ch.) 


Sta. 


Magnetic Bearings 


Distances (ch.) 


A 


N. 12° 15' W. 


33.65 


H 


S. 64° 9' W. 


28.79 


B 


N. 65° 15' E. 


83 


I 


N. 45° 15' W. 


3.07 


C 


S. 13° 30' E. 


27.25 


K 


N. 38° E. 


12.58 


D 


S. 0° 45' E. 


37.95 


L 


N. 23° 45' W. 


6.95 


E 


S. 70° W. 


48.44 


M 


N. 65° 15' E. 


37.55 


F 


S. 74° 43' W. 


45.50 


N 


X. 85° E. 


8.00 


G 


S. 70° 24' W. 


2.80 


O 


N. 14° W. 


4.47 



A ns. 546.84 A. 



Sta. 


Bearings 


Distances (rd.) 


Sta. 


Bearings 


Distances (rd. i 


1 


S. 2°E. 


16.20 


9 


S. 58£° W. 


4.76 


2 


S. 55£°E. 


6.24 


10 


S. 86|°W. 


17.08 


3 


S. 6° W. 


5.32 


11 


S. 86° W. 


41.76 


4 


S. 84£°W. 


14.72 


12 


X.lfl^W. 


16.48 


5 


S. 9°E. 


3.12 


13 


X.7s.', K. 


16.80 


6 


S. 79£° W. 


10.16 


14 


N.lli°W. 


22 


7 


S. 7fW. 


13.12 


15 


N.55J°E. 


18.52 


8 


S. 38£°W. 


9.80 


16 


N. 84° E. 


64.96 



106 



PLANE SURVEYING 



[Art. 169 



Sta. 


Bearings 


Distances 
(ft.) 


A 


N. 68° E. 


137 


B 


S. 17°28'E. 


268 


C 


S. 72° 45' W. 


127.5 


D 


N. 19° 30' W. 


258 



Sta. 


Bearings 


Distances 
(rd.) 


1 
2 
3 
4 


N. 49° 10' W. 
N. 18° 25' E. 
S. 49° 10' E. 
S. 18° 25' W. 


12.50 
17.95 
12.50 
17.95 



Sta. 


Bearings 


Distances 
(poles) 


Sta. 


Bearings 


Distances 

(poles) 


A 
B 
C 
D 
E 


N. 52i° E. 
N. 72° E. 
N. 51° E. 
N. 40° E. 
N. 47° E. 


56 
6.68 
4.84 

13.24 

19 


F 
G 
H 

I 
K 


N.29°E. 
N. 21° W. 
N. 55° W. 
S. 36° W. 

S. 61° E. 


19.36 

3.84 
297.92 
147.67 
279.24 



10 



Stations 


Magnetic Bearings 


Distances (ch.) 


A 


N.24°10'W. 


5.19 


B 


S. 88°15'E. 


2.06 


C 


S. 52°15'E. 


1.65 


D 


S. 20° 15' E. 


1.98 


£ 


S. 47° W. 


2.64 



11. A field is bounded as follows : (1) N. 0° 30' W., 30.24 poles ; 
(2) N. 66±° E., 62 poles ; (3) S. 28° 30' E., 41.12 poles ; (4) S. 67° W., 
17.56 poles; (5) S. 14° E., 23 poles; (6) notes omitted. Supply 
the bearing and length of the missing course, and compute the area 
of the field. 

12. In some convenient field, drive a stake to represent the 
point " A," and starting at this station run out the boundary of a 
field, of which the following are the field notes : 



Stations 


Magnetic Bearings 


Distances 
(ft.) 


Station 


Magnetic Bearings 


Distances 
(ft.) 


A 


S. 80° 15' E. 


310 


D 


N. 79° 20' W. 


221.8 


B 


S. 21° 19' W. 


225.6 


£ 


N. 2° 10' E. 


345.4 


C 


S. 47° 22' W. 


290.7 


F 


N. 69° 53' E. 


205.5 



Note. — This is a good exercise for the student. 
done in running old lines (see Chapter III). 



It is just what often has to be 



Art. 169] COMPUTATION OF AREAS 107 

13. Suppose that when the field notes in Example 12 were taken, 
the declination of the needle was 4° 30' E., but now the declination 
is 2° 45' E., a difference of 1° 45'. Change all the bearings by this 
amount, and beginning at the same stake as in Example 12, run out 
the boundary, using the changed bearings, and plant stakes at each 
corner. Compare the new positions of the corners with the old. 
Has the area been changed ? 

14. With the needle-compass run a line 40 chains long, N. 20° E. ; 
starting at the same point run a line 40 chains, N. 20° 10' E., and 
measure the distance between the ends of the two lines. 



CHAPTER V 

TRANSIT SURVEYING* 



I. FIELD WORK AND COMPUTATION OF AREAS 

170. In Chapter I we have learned something about the use of a 
transit. In city and railroad surveying the transit is the favorite 
instrument, as the needle-compass and the solar compass are too 
uncertain where precision is required. In regions where land is 
valuable, especially in the proximity of cities, the transit is now 
often used for farm surveying. 

171. Azimuth. — The true azimuth of a line is the angle which 
it makes with the geographic (or true) meridian (compare Art. 20). 

The azimuth of a line with reference to any given line AB, as 
a reference line (e.g., some preceding line of a survey), is the angle 

made by the line with 
AB (not BA), or AB 
prolonged, the measure- 



ment being always around 
to the right (clockwise 
motion) from 0° to 360°. 
If the reference line 
is a meridian, then the 
bearings of the courses 
can be at once deduced 
from their azimuths. For 
example, suppose AB, the 
line with reference to 
which azimuths are to be 
taken, coincides with the 
meridian, and the azimuth of AC is 40°; of AD, 145°; of AE, 240°; 

* The reader should remember that by "Transit Surveying" is meant surveying 
with a transit, and that there is no vital difference in principle between " Transit Sur- 
veying," so called, and "compass surveying," for example. 

108 




Art. 173] 



TRANSIT SURVEYING 



109 




of AF, 330°; then it is evident that the bearing of AC is N. 40° E.; 
AD, S. 35° E. ; AE, S. 60° W. ; and AF, N. 30° W. Of course azi- 
muth ol Ae = 90°, of As= 180°, of Aw = 
270°, and of An = 360° or 0°. 

172. In this book we shall reckon 
azimuths in accordance with the definition 
given in the last article ; that is, if the 
reference line is a meridian, azimuth will 
be counted from the north point,* clock- 
wise. Thus, the azimuth of BC, in figure 
below, with AB is the angle xBC; the 
azimuth of CD with AB is the angle yCD, 
zy being parallel to AB. 

The field operation of getting azimuths 
is given in the next article. 

173. To survey a Farm with the Transit. — Suppose we wish to 
refer all lines of the field ABCDEFdA (Fig. 59) to the line AB. 
Set up the transit so that the plummet is directly over the point B 
(marked by a tack in the top of a stake), clamp the alidade to the 
limb so that one of the verniers, say vernier A, shall read zero, and 
sight the point A, then clamp the limb to the spindle, getting the 
line of sight exactly on the point A by means of the tangent screw 
belonging to the lower limb. Revolve the telescope on its H (hori- 
zontal) axis, and it will then point in the direction Bp, the prolonga- 
tion of AB. Loosen the alidade and sight C, or some well-defined 
point on the next course BC, bisecting the point exactly by means of 
the alidade tangent screw, "\ the reading of the vernier, in this case 
82° 33', gives the angle pBC, the azimuth of BC with. AB. After 
reading this angle and recording it, loosen the limb, take up the 
transit and set it over C (as the party moves along, the courses should 
be carefully measured with a chain or tape and their lengths recorded) ; 
revolve the telescope back on its ^T-axis (if this has not already been 
done), sight B, and clamp the limb, using the lower tangent screw 
to bisect the point (remember that now both the alidade and limb 
are clamped) ; then revolve the telescope about its ^T-axis, making it 

* Astronomers and geodesists reckon azimuth around from the south point, clock- 
wise. Professor Johnson, in his work on Surveying already mentioned, thus reckons 
it, while Professor Raymond finds it more convenient to reckon it from the north point. 

t The beginner is cautioned against the common mistake of turning the wrong tan- 
gent screw. 



110 



PLANE SURVEYING 



[Art. 173 



point towards q ; loosen the alidade and sight the next point D; the 
reading of the vernier, 174° 36', will now give the azimuth of CD 
with AB, or the angle xCD. Notice that after moving the instru- 
ment to C and sighting B, the (horizontal) limb has now its zero- 
point in the direction of xy, or pA,* Also notice that before we 
loosen the alidade the reading of the vernier is still the angle 
pBQ — xCq, and when, after revolving the telescope and loosening 
the alidade, we sight D, we add to this angle the angle q CD, that is, 

xCq + qCD = xCD. 

Lines drawn through the corners parallel to the initial line AB, 
such as yx, give the direction of the zero of the limb at each comer 

before the alidade is loosened. 

Next, after having recorded 
the azimuth of CD, loosen the 
limb and move the instrument 
to D, revolve the telescope back 
on its iST-axis, sight C, using 

^ s lower clamp and tangent screw, 

revolve the telescope on its 
if-axis (toward r), loosen ali- 
dade and sight U, the reading, 
83° 50', will be the azimuth of 
DE. Continue thus till A is 
reached, duly recording all azi- 
muths and distances in the 
proper columns. It is not 
necessary to set up the instru- 
ment at A, but it should always 
be done if possible, for it fur- 
nishes a strong check on the 
measurement of the angles. 
After setting up at J., ori- 
enting the transit as at the other stations, and sighting B (where 
the transit was first set up), the vernier should read 360° or 0°. If 
it does not, there is some mistake in getting the angles, and if the 
error is considerable, the lines should be re-run. In order that this 
last reading shall be 360°, the plummet must be exactly over the 
point, the rod must in every case be bisected by the line of sight while 
it is held exactly over the station sighted, and the alidade must not 

* Getting it in this position is called orienting the instrument. 




Art. 175] 



TRANSIT SURVEYING 



111 



slip.* In the example here given this test reading, from A to B, 
was actually 359° 56 1 ', four minutes out. 

The magnetic bearing of each line should be taken as a rough 
check on the reading of the azimuths, even when not needed for use 
in the deed to the land. Much labor is often saved by thus discover- 
ing, before the instrument is moved, an error in reading an azimuth, 
or possibly an error due to the slipping of the alidade or limb. Using 
the magnetic bearings, the surveyor mentally calculates the angle 
between two adjacent courses and compares this with the same angle 
derived from the azimuths. If they differ much, the error is probably 
in the azimuth. By this means gross errors in the use of the transit 
may be detected. 

174. In his field book, the surveyor records the stations, distances, 
magnetic bearings, and azimuths, as given in the first four columns 
of the form below. On the right-hand page of his field book he will 
naturally give all data concerning location of corners, offsets, etc. 

Form (a). 



o 

GO 


Distances 
(ft.) 


Magnetic 
Bearings 


Azimuths 
with 
AB 


Bearings 

with ab as 
North 


Computed 

Magnetic 
Bearings t 


A 


594.9 


N. 8° 40' E. 


359°56' 



North 


K 8° 13' E. 


B 


294 


S. 88° 55' E. 


82° 33' 


N". 82° 33' E. 


S. 89° 14' E. 


C 


232.8 


S. 3° W. 


174° 36' 


S. 5° 24' E. 


S. 2° 49' W. 


D 


163 


S. 88° E. 


83° 50' 


N". 83° 50' E. 


S. 87° 57' E. 


E 


380 


S. 0° 5' W. 


171° 52' 


S. 8° 8' E. 


S. 0° 5' W. 


F 


337.8 


S. 51° 25' W. 


222° 59' 


S. 42° 59' W. 


S. 51° 12' W. 


G 


362 


N. 47° 25' W. 


304° 11' 


N. 55° 49' W. 


N. 47° 36' W. 



175. Azimuths changed into Bearings. — To obtain the area by the 
D. M. D. method, we change azimuths into bearings. It is convenient 
to get first the bearings of the course, considering AB as a meridian 
(even though it does not run north and south), as given in column 
5, Form (a). This would be sufficient for determining the areas, 
but, as the magnetic bearings are usually required in all forms of 
deeds to farms, we change the bearings by the number of degrees 

* The clamp screws should be made tight, but not too tight. The beginner is apt 
to exert too much force in turning the screws. 

t Assuming the observed bearing of EF, S. 0° 5' W. , as correct. 



112 



PLANE SURVEYING 



[Art. 175 



representing the angle between AB and the magnetic (or true *) 
meridian, so that now all the courses have their real bearings with 
the magnetic (or true) meridian. These new bearings (column 6, 
Form (#)) are called the "computed magnetic bearings." For this 
purpose, we should get with great care the magnetic bearing of AB 
and derive the magnetic bearings of the other courses from their 
bearings with AB (column 5). It may, in some cases, be better to 
assume the observed magnetic bearing of some other course as cor- 
rect, and derive the others from this, using of course their bearings 
with AB. 

In the example we use for illustration, the observed magnetic 
bearing of EF (S. 0° 5' W.) was considered the most reliable, as the 
needle seemed more or less affected by telephone wires and wire 
fences at all the other corners, and could not be relied upon. 

176. Computation of Areas. — After getting the bearings that we 
wish to use, the computation of areas is made exactly as in Compass 
Surveying (see Chapter IV). Far less error is to be expected than 
in compass work ; and as the error may be assumed to be due mainly 
to the chaining, we here use Rule II (Art. 153) in balancing the 
latitudes and departures. 

In transit surveys, where the angle may be read to within two 
minutes, a traverse table cannot be used advantageously. Most 
surveyors will prefer to compute each latitude and departure by 
logarithms, as was done in this example. Below we give a conven- 
ient form for arranging the logarithms (Form (b)). Here a five-place 
table was used, but in most cases a four-place table will prove suffi- 
ciently accurate. The reduction of the area is given in Form (c), 
page 113. 

Form (b) 



Courses 


AB 


BC 


CD 


BE 


EF 


FG 


GA 


log sin B 


9.15508 


9.99996 


8.69144 


9.99972 


7.16270 


9.89173 


9.86832 


log dist. 


2.77444 


2.46835 


2.36698 


2.21219 


2.57978 


2.52866 


2.55871 


log cos B 


9.99552 


8.11693 


9.99948 


8.55354 


0.00000 


9.79699 


9.82885 


logD. 


1.92952 


2.46831 


1.05842 


2.21191 


9.74248 


2.42039 


2.42703 


D. 


85.02 


293.97 


11.44 


162.90 


0.55 


263.26 


267.32 


logL. 


2.76996 


0.58528 


2.36646 


0.76573 


2.57978 


2.32565 


2.38756 


L. 


588.79 


3.85 


232.52 


5.83 


380.0 


211.66 


244.09 



* If a solar attachment is used, the bearings would be referred to the true 
meridian. 



Art. 176] 



TRANSIT SURVEYING 



113 



Q £ PQ 



Tfl <n co o 

o co co oo 

<N <M i-H CO 



£ £ 



co 


CO 


03 


co 


iC 


1 — 1 


r- 


CD 


CO 


CO 


CO 


00 


o 


CO 



no 


co 


CO 




CO 


o 


CO 

co 


CD 
CM 







1 


CO 


03 


CO 
CO 


o 
o 


CO 
CO 


CO 
CO 


CO 
CO 


EC 


co 


co 

0} 


iO 


o 

CO 

co 


(N 


co 

CO 

co 


co 
co 



> 



<1 PQ O P H 



114 



PLANE SURVEYING 



[Art. 177 



177. By the method just given, let the student compute the 
area of a lot of which the field notes are as follows : 

" Drill Field " 



Stations 


Distances (ch.) 


Azimuths with AB 


Magnetic Bearings* 


A 


4.41 





N. 22° 40' W. 


B 


24.82 


90° 6' 


N. 67° 30' E. 


C 


4.43 


180° 2' 


S. 22° 25' E. 


D 


3.38 


267° 29' 


S. 65° 30' W. 


E 


2.29 


261° 8' 


S. 59° W. 


F 


19.20 


271° 40' 


S. 69° 40' W. 



178. Interior Angles. — Another way of using the transit in a 
farm survey is to measure the interior angles, from which the azi- 
muths or bearings with any of the courses may be readily derived. 
This was done in the following survey : 



Field Notes 



Stations 


Distances (ft.) 


Magnetic Bearings 


Interior Angles 


A 


289.1 


N. 51° 25' E. 


270° 14' 


B 


776.4 




48° 48' 


C 


402.0 




49° 54' ' 


D 


426.3 




173° 9' 


E 


298.3 




89° 36' 


F 


246.0 


S. 38° 30' E. 


88° 19' 



In the above table the interior angle recorded opposite A means 
that angle FAB = 270° 14', and ABO '= 48° 48', etc. It is left as an 
exercise for the student to compute from the interior angles the 
azimuths with respect to AB, and the magnetic bearings of all the 
courses, considering that of AB, N. 51° 25' E., as correct. Also plot 
and compute the area. 

179. Deflection Angles. — Another way of getting the direction 
of lines is to measure the deflection angles, a deflection angle being 
the angle which a line of a traverse makes with the prolongation of 
the line immediately preceding it ; if it passes to the right of the 

* In getting the "computed magnetic bearings," assume the bearing of AB, 
N. 22° 40' W., as correct. The error of closure will be found to be very small. 



Art. 180] 



TRANSIT SURVEYING 



115 



line prolonged, R. is written after the angle, if to the left, L. 
is a convenient method to employ in running a traverse 
of a highway or railroad. 

For example, suppose that a section of an irregular 
highway from A to Gr, Fig. 60, is to be traversed, 
and that it is found convenient to make stations at 
A, B, C, D, E, F, and Gr, the exact bounds of the road 
being obtained by measuring offsets from the traverse i 
line. The instrument is set up at B, and, the vernier 
of the alidade being set at zero, A is sighted, then the 
telescope is revolved on its iZ"-axis, and the angle 
tBC ( = 46°) is obtained in the usual way. It is writ- 
ten 46° L., to indicate that the deflection is to the left. 
Measure the distances, AB, BC, etc., as the work pro- 
gresses, and take offsets at intervals sufficiently close 
to determine the boundary of the road, if that is 
required. Move the instrument to C, again set the 
vernier at zero, and as before measure the deflection 
s(72)= 75° 40' R., noting that this time the deflection 
is to the right. Proceed in this way till the end of 
the traverse is reached. It is often an advantage to 
have some of the lines entirely outside the limits of 
the road, the centre line of which is readily deter- 
mined by offsets. 



This 



Fig. 60 



180. The length of each course may be recorded 
separately, but in work where deflection angles are 
used, the lines are usually measured continuously from the begin- 
ning, and the stations are indicated, not by letters, but by numerals 
which indicate their distance from some initial point ; thus : 

" " = the initial station. 
" 1 " = station 100 ft. from the initial point. 
" 2 " = station 200 ft. from the initial point. 
" 5 4- 40 " = station 540 ft. from the initial point. 



This method of naming the stations has the advantage of show- 
ing at a glance the distance that has been covered. 

Following we give, as an illustration, notes actually taken in 
traversing a road : 



116 



PLANE SURVEYING 



[Art. 180 



Notes. — Road to Green's View 



Stations 
(Total Dis- 
tances) 


Deflection 
Angles 


Remarks 


A 


58° 15' R. 


A"0" is also A"0" of preceding traverse {abed . . .), 


A 1 + 69 


42° 50' L. 


and the angle 58° 15' R. is the deflection with ba of 


A 10 + 87 


11°45'R. 


that traverse. 


A 14 + 36 


46°43'R. 


A " 1 + 69 " = Hub east side of road, 2 ft. from fence 


A 19 + 49 


60° 12' L. 


post, and on line with K-S's south fence. 


A 22 + 23 


30°27 / L. 


A " 10 + 87 " = Hub at forks of road, Milhado's corner. 


A 29 + 50 


9°25'L. 


A " 19 + 49 " = Hub near old log at forks of roads to 


A 32 + 54 


10° 7'L. 


Green's View and Hodgson's Spring. 


A 37 + 93 


13° 12' L. 


A " 29 + 50 " = Hub on right side of road, at fork of 


A 44 + 05 


2°48'L. 


Beck with Point road. Deflection of B. P. road = 


A 46 + 72 


34°52'R. 


38° 39' R. 


A 50 + 98 


— 


A "50 + 98," Hub 26 ft. from road, right of Green's 
View on brow of point. 



Some such form as this is usually employed in running out high- 
ways and railroads. It is beyond the scope of this work to describe 
the methods of laying out curves, though the field operations are 
very simple. 

EXAMPLES 

1. Survey a field * or farm with a transit, recording the azimuths 
of the courses and their magnetic bearings as a check, plot and com- 
pute its area by the method of Art. 176. 

2. Let another party, or the same party, survey the same field, 
finding the interior angles, and plot and compute the area, comparing 
the results with those found in Example 1. 

3. Lay off, with the transit and a chain or tape, a square f to con- 
tain 108,900 sq. ft. 

4. Run a traverse out one road, returning by another route to 
the starting point, by means of deflection angles. Test the accuracy 
of the work by plotting. 

5. By the method of Art. 143, establish a true meridian line, and 
determine the declination of the needle at the point of observation. 

* The student should have as much practice in field work as possible, and he can 
thus make his own examples, and supplement the small list given here. 

f The beginner will be surprised to find that it requires great care to lay off a 

square. 



Art. 182] 



TRANSIT SURVEYING 



117 



Plot, determine the error of closure, and compute the areas in the 
following transit surveys. 



O 

H 

CO 


Azimuths with 

SJST, a True 

Meridian 


Distances 
(ft.) 


A 


267° 22' 


742 


B 


186° 52' 


427.7 


C 


174° 45' 


271.2 


D 


172° 28' 


266.3 


E 


41° 42' 


722.3 


F 


29° 12' 


519.2 



CQ 

S5 

O 


Azimuths 


Magnetic 


Distances 


CO 


with AB 


Bearings 


(ft.) 


A 





N. 39° 15' W. 


408 


B 


91° 13' 


N. 51° 45' E. 


463 


C 


173° 55' 


S. 45° 15' E. 


415.1 


D 


257° 39' 


S. 38° 45' W. 


208.4 


E 


188° 31' 


S. 30° 45' E. 


165.1 


F 


280° 8' 


S. 61° W. 


334.9 


G 


17° 8' 


N. 22° 15' W. 


171.5 



Arts. 6.11 A. 



Station 
to 


Azimuths 
with True 


Distances 
(ch.) 


Station 


Meridian 


1 to 2 


335° 50' 


51.90 


2 to 3 


91° 42' 


20.60 


3 to 4 


127° 46' 


16.50 


4 to 5 


159° 47'. 


• 19.80 


5 to 1 


227° 


26.40 



Stations 


Azimuths 
with AB 


Distances 
(ft.) 


A 





310 


B 


101° 34' 


225.6 


C 


127° 37' 


290.7 


D 


180° 55' 


221.8 


E 


267° 25' 


345.4 


F 


330° 8' 


205.5 



II. LAYING OUT AND DIVIDING LAND 

181. To lay out a lot of given area in the shape of a triangle, 
the base being given. 

Let S = given area, 

AB = b, given base. 

Then the altitude is 

h=OD= M, 
o 

and evidently the vertex may lie anywhere in a line Ox, parallel to 
AB, at the distance h from it. 

182. To lay off a lot of given area, in shape of a triangle, one 
side and an adjacent angle being given. 

Here A = c, angle BA = A are given. 




118 



PLANE SURVEYING 

2S 



[Art. 183 



Then h — c sin A, and the base, b = 



c sin A 



Hence lay off AB 



2S 



e sin A 



and ABO will be the required triangle. 



183. To lay out a lot of given area in shape of an equilateral 
triangle. 

Let x = the side of the triangle. 

Then S = ^ V3 (see Table XII) 



and 



x* = 



4 

V3* 



from which we find x, and the construction of the triangle is easy. 

If the lot is to be in the shape of a square, x = V#. In both 
cases, if S is given in acres, it must first be reduced to square chains, 
or some convenient unit. 

184. To lay off a rectangle of given area, one side, a, being given. 

Here the side perpendicular to the given side is h == — , and the 
construction follows. 



For a parallelogram of given area and base, we have h = 



S 



At the distance h from the base draw a line parallel to the base. 
Lines drawn parallel to each other from the extremities of the base 
intersecting this parallel line determine the parallelogram, there being 
an indefinite number of solutions. 

185. To lay out a lot of given area in the form of a regular 

polygon of any number of sides. 

Given S = area, n = number of 
sides. 

Let x = length of a side, as 
AB in Fig. 62, and y = OA = the 
radius of the circumscribed circle. 




Then 
Now 
AM 



n x OM xAM=S. 
180 c 



OM: 



cot 



Fig. 62 



for angle A OB = 



2 

360 c 
n 



Art. 186] TRANSIT SURVEYING 119 

x 2 180° 
Therefore n— cot = S, 



4 



n 



whence x is derived. 



A • x 180° 
Again y — - cosec 

Having found in this way the length of a side, stake out AB = x, 

360° 
set the transit over B, make the deflection angle xBC = , and lay 

oftBC=x. n 

Move the instrument to C, again deflect by an angle equal to 

360° 

, and lay off CD = x. Continue thus until A is reached again. 

n 

If the field is small and all the corners visible from the centre, 

a better method is to find the centre of the circumscribed circle 

(if the position of one side, say AB, is fixed, let the student show 

how the centre may be determined), at the required distance, y, from 

A and B ; then set up the instrument over the centre, measure the 

360° 

angles BOO, COD, etc. = , and lay off on the lines thus deter- 

n 

mined the distances OC ' — OD = OE, etc. = y. 

If the problem is to lay off a circle of given area, the radius is 
determined by the equation S = irx 2 . The circumference can be laid 
off by the method of this article. Take n as large as is convenient, 
thus establishing as many points as possible, all of which will lie on 
the circle of radius x. 

186. To lay out a lot of given area in the form of an ellipse, the 
greater and lesser diameters to be in a given ratio m : n. 

Let #=area, 2 mx — greater diam- 
eter, 2 nx = lesser diameter. 

Now the area of an ellipse is irab, 
where a and b represent the semi- 
diameters (Table XII). 




Hence irmnx 2 = S, .-. x = ^/S-r-irmn, 
and thence mx and nx. 

Construction. — A small ellipse 
can be conveniently laid off as fol- 
lows : 

Suppose its greater diameter = 2 a, its lesser diameter = 2b. 

Measure AA'=2a, and from its centre C lay off CF= OF' = 
Va 2 - b 2 . Fix the ends of a chain or wire (some material sufficiently 
flexible, but not easily stretched) of length 2 a at F and F, and with 




Fig. 64 



120 PLANE SURVEYING [Art. 187 

a continuous motion of a marking pin P keep the wire taut ; the pin 
will trace out the ellipse. An ellipse can be traced on paper in this 
way, but care must be taken to prevent slipping or the stretching of 
the guiding string. 

187. To divide a given triangle into two parts in the ratio oimin 
by a line parallel to one side. 

Let ABC be the given triangle. 

Denote the sides opposite A, B, C by a, 
5, c respectively. The problem is to draw 
PR parallel to AB, so that area ABRP : PRO 
= n:m. Let CP = x, PR = y. Then, since 
areas of similar plane figures are to each 
other as the squares of homologous sides, we have 

area ABC: area PRC=b 2 : x\ 

but area A B C: area PR C— m + n : m. 

,\ b 2 : x 2 = m + n : m. 

.-. x 2 = b 2 , or x = b\- — . 

m + n *m + n 

Hence, measure CP = #, set up the instrument at P, lay off the 
angle CPR = A, and range out the line PR. As a check on the 
work, measure PR, and compare, the result with its length deter- 
mined by the proportion : PR : AB= CP : CA, or y : c = x : b . 

If the triangle is to be divided into equal parts, x— %bV2 and 
y=lcV2. 

188. To divide a given triangle into 
two parts in the ratio of m : n by a line 
from a vertex to the opposite side. 

To draw CP, so that ACP:PCB=m:n. 

Let AP — x. Then, since the triangles 
ACP and ACB have the same altitude, 
they are to each other as their bases, 

ACP : ACB =AP:AB = x:c, 

but ACP:ACB = m:m + n. 

mc 
.'. x : c = m : m 4- n, or x = 




m+ n 



If m — n, x = -. 

2 



Art. 190] 



TRANSIT SURVEYING 



121 



189. To divide a given quadrilateral into two parts having a 
given ratio, by a line extending from a given point in one of the 
sides. 

Suppose it is required 
to divide the quadrilateral 
ABCD into two parts, 8, 
8', so that 8 : 8' = m : n, by 
a line drawn from P, a 
point in DA. 

Make a plot of the 
quadrilateral on as large a scale as 
practicable. Through P draw a trial 
line PQ. Measure carefully by scale 
the lines PQ, BQ, and AQ, and com- 
pute the area of ABQP. Compare it 



m 




It will 



Fig. 66 



with the value of S — —8. 

n 

probably differ from 8 considerably ; call this difference x. Now 

draw a perpendicular p = PE, from P to BO, and get its length 

by the scale. Then the base of a triangle with vertex at P, 

2 x 
which is to be added to or subtracted from ABQP, is = — . Lay 



2x 



P 



off from Q on BO this distance QR= — , and ABRP will be the 

P 
required area. In any case BR and PR should be ranged out on 

the ground and measured. If the area is not exact enough, another 

trial may make it all right. 

190. Given a Polygon, to divide it into Two Parts having a Given 
Ratio. — This is a problem of which the preceding is a special case. 
It is perfectly general, and the method may be readily applied where 
an estate, or a single field, given by the bearings and distances of the 
courses, is to be divided into two parts having a given ratio, or where 
a given area is to be cut off. An example will make the process clear. 

For illustration, take "Elliott Park," a plot of which is given on 
page 110, Fig. 59, and the computation of the area on page 113 ; 
and suppose it is required to cut off from the south end of the park, 
by a line drawn from F, two acres, or 87,120 sq. ft. 

We first calculate the area of the triangle FGA (see Case I, Art. 
160), and find it to be 61,640 sq. ft. Subtracting this from 87,120, 
we get 25,480 sq. ft. Now, having carefully plotted the field (Fig. 
59), we measure by our scale the perpendicular Fn, dropped from F 



122 



PLANE SURVEYING 



[Art. 190 



to AB, finding it to be 532 ft. Dividing double the additional area 
(2 x 25480) required by 532 we get 95.8 for the distance AR, and 
ARFOA is the required area. Test this both on the plot and in the 
field by finding the bearing and length of the course FR. 

The above problems in laying out and dividing land are merely 
suggestive. The surveyor's knowledge of geometry and trigonom- 
etry, as well as his personal experience in the field, will suggest 
many other similar ones. 




Fig. 67 



III. STADIA SURVEYING* 

191. The geometrical theorem that in similar triangles homolo- 
gous sides are proportional furnishes the fundamental principle upon 

which stadia measurements 
are based. 

Suppose BE Q = a) and 
BO (= a') to be parallel, 
and let their distances 
from the vertex A be d 
and d' respectively ; then 
a' : a = d' : d, that is, the lengths of parallel lines subtending an 
angle are proportional to their distances from the vertex ; for exam- 
ple, if a' = 2 a, then d' = 2 d. 

192. Stadia Wires. — The practical application of this principle is 
accomplished by having in the reticule of the telescope two hori- 
zontal wires (see Art. 62). The lengths intercepted by these parallel 
wires on a vertical rod held in front of the instrument are, with a 
slight modification, proportional to the distances of the rod from the 
instrument. The theory of stadia measurements, which we shall 
presently establish, will show what this modification is. These wires 
may be fitted to any telescope, but are especially useful in the tele- 
scope of the transit and the alidade of the plane-table. 

193. Horizontal Sights. — In Fig. 68, LU is supposed to be the 
objective lens, of which F 1 and FJ 2 are the " principal points," f and 
F is the principal focus, or the position of the image for an object 

* In preparing this account of Stadia Surveying, the author has consulted articles 
by George J. Specht and Arthur Winslow, both published by Messrs. D. Van Nostrand 
Co. 

t For simplicity these points are usually made to coincide at the centre of the lens, 
O, which, while not rigidly accurate, makes no appreciable difference in the results 
obtained. 



Art. 194] 



TRANSIT SURVEYING 



123 



an infinite distance away, and is the centre of the telescope over 
the plummet. 

Let AB = s = portion of rod intercepted between the stadia wires, 
and A'B' = i= the image of AB at the reticule. 



Let 



E^F '=/, the principal focal length, 



EJ=M 



\ conjugate foci. 



E 2 P=f 2 ) 
E 2 C= c, distance from objective to centre of the instrument. 




Fig. 68 



From similar triangles, AE 2 B, A*E X B\ we have 
E 2 P: E 1 I=AB:A'B', or 

f 2 i/j = s : i. 

From a law of lenses, — | — = - . 

A ft f 

Eliminating f x between (1) and (2), we have 



and putting 






d=CP 

f. 



(1) 
(2) 

(3) 

(4) 



i+f+c. 

We see from this that ^s is the distance from F', a point in front 

i 

of the objective at a distance from it equal to the focal length. As 
we want the distance from the centre of the instrument, we must add 
to this the constant, f + c. 

194. It is the necessity of adding this constant to the reading 
from the stadia rod that has prejudiced many surveyors against 
stadia work. To obviate this difficulty, some have the rod arbi- 



124 



PLANE SURVEYING 



[Art. 194 




Fig. 69 



trarily graduated, so that, at the distance of an average sight, say 
800 ft., the same number of units of graduation are intercepted on 

the rod as units of 
length are contained 
in the distance. It 
should be remem- 
bered, however, that 
the distance read is 
strictly correct only 
for the 300 ft. 

The stadia wires 
are made either fixed 
or adjustable. If 
fixed, and this is usually the better plan, it is convenient to space 
them so that k = 100 ; then, if the rod reading is 2 ft., the distance 
is 200 ft. plus the constant /+ c, or, if the rod reading is 3.4, the 
distance is 340 ft. plus /+ c. 

195. This constant, which in ordinary transits varies from 10 to 
16 in., should be determined for the instrument used, as follows : 
Measure with a rule the distance from the objective to the centre of 
the instrument, which gives c. Focus the telescope on a very distant 
object, preferably the moon or a star, and measure the distance from 
the objective to the reticule, which will be /. When an instrument 
with fixed stadia wires is purchased, the maker will give the value 
of this constant and also Jc. 

In the 11-in. telescopes of Gurley, c— 5yV' and /= 8", f+c 

196. The value of h may be determined as follows : 

Set up the instrument and measure off in front of the plummet a 
distance =f-\-c. Then from this point, which we will call A, meas- 
ure some convenient distance, as 400 ft., to a point B in the line of 
sight. Hold the rod vertically at B and read the space intercepted 
by the wires, calling this s' ; then 

4oa+/+-<> = *«'+/-+*, 

.-. Jcs r = 400, 



or 



7c = — r , and h is known. 



If 



s' = 4, k= 100. 



Art. 197] TRAXSIT SURVEYING 125 

Equation (4), Art. 193, gives the horizontal distance on level or 
nearly level ground. As most lines are run on ground that is not 
level, we need another formula. 

Formulae for Inclined Sights 

197. In going up or down a hill, if the rod is held perpendicular 
to the line of sight, Formula (4) will give the linear distance to the 
rod. But it is inconvenient to hold the rod in this inclined position, 
and besides, this linear distance would have to be reduced to horizon- 
tal measurement. The formula that we shall now deduce gives the 
horizontal distance when the stadia rod is held in a vertical position. 

Let C, Fig. 69, be the centre of the telescope, over the point IT, 
EF the stadia rod held vertically at the point E, CP the line of 
sight, and AB the part intercepted on the stadia rod. 

Draw through P a line ab perpendicular to CP. 

Let AB = s, ab = s', angle PCD=v = bPB = aPA. Now the 
angle ACB is so small that no appreciable error will be made if we 
consider the angles BbP and AaP as right angles. Assuming this, 
we have ap = Ap cog ^ ph = pp cQg y ^ 

.*. ab = AB cos v, or $' = s cos v. 

Now, by Formula (4), putting s r for s, 

CP = ks f + (f+c), (5) 

.-. CP = ks cos v +f+c, (6) 

and CD — d = CP cos v, 

or d = ks cos 2 v + (/ + c) cos v, (7) 

which is the required formula. 

We also have for the elevation of the point E above H (provided 
that P is at a distance from the ground at E equal to the height of 
the instrument), 

PD = h = CP sin v, 

hence h = ks cos v sin v + (/ + <?) sin v, 

or h = | ks sin 2 v -f (/ + c) sin v. (8) 

Hence, Formula (7) gives the distance, and (8) the elevation, 
reading v on the vertical arc of the instrument, and s on the rod, 
&, /, and c being constant for the instrument used. 



126 PLANE SURVEYING [Art. 198 

198. Stadia Tables. — To save the labor of solving Equations (7) 
and (8) every time a reading is taken, tables are constructed giving 
distances and heights * for observed values of v and s. For an illus- 
tration, see Table XVI, in which d and h are computed from the for- 
mula for a stadia reading of 100 ft. (or metres), with angles up to 30°. 

The use of this table involves one multiplication and one addi- 
tion. For instance, if we want the horizontal distance (Jl) and 
difference of elevation (c?) corresponding to 8= 3.42, v = 6° 30', we 
get the proper values from the table, page 156, in the column headed 
6°, opposite 30'. These values must be multiplied by 3.42, and to 
the results must be added the corrections, (c +/) cos v and (<?+/) 
sin v, respectively, also given in the same column opposite the letter 
c (say 1.00), thus, 

a = 98.72 x 3.42 + 0.99 = 338.61, 

h = 11.25 x 3.42 + 0.11= 38.58. 

Three values of c are given, to enable the surveyor to use a value 
which nearly corresponds to the particular telescope that he is using. 

It is not necessary to use the table for getting the horizontal dis- 
tance d for angles of elevation or depression less than about 4°, and if 
an error of 1 in 100 is permissible, then the reduction need not be 
used under 6°. (In such cases Formula (4) gives the distance.) If 
c +/ be neglected, as well as v, these two errors tend to compensate 
for each other. 

In obtaining the difference of elevation h, the term in c+/ may 
be omitted for angles less than 6°, if errors of 0.1 ft. are unim- 
portant. 

199. The chief advantage of stadia work lies in the fact that it 
saves the labor of chaining. This advantage is especially felt in 
rough, rolling country. As to its accuracy, the combined experience 
of many surveyors seems to show that it is as accurate on fairly 
level ground as ordinarily good chaining, more accurate than un- 
usually good chaining on rolling ground, and much more accurate 
on any ground than ordinary chaining. Care must be taken to 
avoid gross errors. The most common errors appear to be compen- 
sating rather than cumulative, as is shown in the example given in 
the following article, f 

* A particular form of the slide-rule is made for reading with ease and rapidity 
the distance and heights. 

t Taken, by permission, from Professor J. B. Johnson's "Theory and Practice of 
Surveying." 



Art. 200] 



TRANSIT SURVEYING 



127 



200. "A good example of the use of the transit and stadia method 
in running levels in city topographic surveys is found in the recent 
topographic survey of St. Louis. In this survey a transit and stadia 

Results of Leveling by the Stadia Method 





Azimuth 
Errors 


Accumulated 


Distances in 


Errors in 


Error of Closure 
in Elevation 


Stations 


Errors in 


Miles from 


Horizontal 




Elevation (ft.) 


Starting-point 


Measurement 


between Check- 
points (ft.) 


2547 


o / n 

1 20 


+ 0.42 


2.0 


+ 1 : 387 


0.42 


2803 


50 


+ 0.46 


4.1 


+ 1:619 


0.04 


2777 


2 20 


+ 0.17 


6.2 


+ 1 : 1177 


0.29 


1332 


2 00 


+ 0.09 


7.8 


+ 1 : 1149 


0.08 


1393 


2 00 


+ 0.50 


9.2 


+ 1 : 987 


0.41 


774 


3 33 


+ 0.52 


10.9 


+ 1 : 1000 


0.02 


400 


8 13 


+ 0.09 


12.3 


+ 1 : 1084 


0.43 


389 


7 06 


+ 0.08 


14.6 


+ 1 : 1203 


0.01 


1839 


9 32 


+ 0.25 


16.3 


+ 1 : 1025 


0.17 


1871 


9 52 


-0.01 


18.4 


+ 1 : 965 


0.26 


2008 


10 42 


+ 0.14 


20.5 


+ 1 : 836 


0.15 


2067 


11 42 


+ 0.37 


22.4 


+ 1 : 877 


0.23 


41 


12 12 


+ 0.39 


23.8 


+ 1 : 961 


0.02 


2292 


11 42 


+ 0.77 


25.2 


+ 1 : 1063 


0.38 


2304 


10 42 


+ 1.37 


27.0 


+ 1 : 1139 


0.60 


1699 


10 42 


+ 1.00 


28.9 


+ 1 : 1484 


0.37 


566 


9 40 


+ 1.23 


30.3 


+ 1 : 1644 


0.23 


1500 


9 05 


+ 1.03 


31.8 


+ 1 : 1724 


0.20 


1488 


10 00 


+ 0.68 


33.5 


+ 1 : 2267 


0.35 


937 


12 35 


+ 0.94 


34.9 


+ 1 : 3291 


0.26 


958 


9 18 


+ 0.98 


36.2 


+ 1 : 3945 


0.04 


1115 


9 30 


+ 0.64 


37.9 


+ 1 : 5174 


0.34 


2476 


8 20 


+ 0.36 


39.8 


+ 1 : 6420 


0.28 


124 


8 20 


+ 0.64 


40.4 


+ 1 : 6332 


0.28 



line was run over 40 mi. long. At twenty-four points along this 
circuit the line checked on triangulation points and precise-level 
bench-marks, with the results shown in the table above. 

" The average length of the lines between check-points was 1.7 mi. 
and the average error for this distance was 0.24 of a foot, or 0.18 of 
a foot per mile of line. 

" It should be noted that while the total accumulated error in ele- 
vation for the entire 40 mi. was but 0.64 of a foot, at a point on the 



line distant 20 mi. from 



the beginning 



the error in elevation was 



128 PLANE SURVEYING [Art. 201 

zero, while in 7 mi. more it was over twice the error at the end of 
the circuit, thus emphasizing the fact that the errors in such work 
tend to compensate." 

201. Stadia Rods. — The ordinary leveling-rod, with target or 
self-reading, may be used, but it is generally more convenient to 
have a special rod constructed for the purpose. Many designs are in 
use, the main object being to get a form which may be easily read 
at a distance. Such a self -reading rod is described in Art. 98, and 
represented in Fig. 22. 

202. To any one who has had experience in handling a transit, 
the field operations in stadia work will present no special difficulties. 
The surveyor must be careful not to read the space on the rod inter- 
cepted between one stadia wire and the middle horizontal wire instead 
of the space (which is twice as great) between the two stadia wires. 
In the next chapter attention will be called to the use of the stadia 
in topographical work. In getting elevations, remember that the 
middle wire should (to be exact) cut the rod at a distance above the 
ground equal to the height of the instrument. For the best results 
a telescope of good defining power is desirable, and the instrument 
must be in nearly perfect adjustment. 

203. Field Notes. — The form of the field notes varies according 
to the object of the survey. 

When a traverse is run and at the same time the profile of the 
traverse is desired, the form for field notes and reduction given below 
is suggested. These notes were taken in running a line for a highway. 

As the vertical angle v was small on this part of the line, the 
example selected serves to show that, if the distance (t?) alone is 
desired, the reduction formulae for inclined sights may be disre- 
garded for small values of v, as has already been stated. Here the 
stadia station* G- is itself used as a B. M., and is 100 ft. above a 
certain datum plane (see Art. 218). The last reading from \DGr is 
on Eli/", which is used as a T. P.,f its elevation being found to be 
120. The transit is then moved to IT. The numbers given in the 
h column are now the elevations above or below □ iT", and are to be 
added (algebraically) to 120 to obtain the final elevations given in 
the 7th column. Here □ H is evidently on the line HI, and is used 
as an intermediate station to prolong the line, and incidentally as a 
T. P. The next station, 2, is also used as a T. P. 

* Denoted by Q] G. t Turning-point ; see Art. 217. 






Art. 204] 



TRANSIT SURVEYING 



129 



In the form, □ H and B I could have been written in the line just 
above where they are placed; but the present arrangement avoids 
confusion. If the work is to be continued, the last reading should 
be on a B. M. 

Note. — If JT] H was used simply as a T. P. , and was not necessary for prolonging 
the line, time could be saved by occupying fT] /, on moving from B #, and taking back- 
sights on H and the points between H and / ; but in considering the sign of v, it must 
be remembered that the sight has been reversed, and this must be indicated in the 
held book. 

In the reduction below, the c of the table is supposed to have the 
value 1. 

Form for Stadia Notes 



Stations 


Azimuths* 


V 


s 


d 


h 


Elevations 


Total 
Distances 


BO 


90° 








0.0 


100.0 


0.00 






+ 2° 20' 


0.83 


83.86 


+ 3.4 


103.4 


83.86 






+ 1° 45' 


1.50 


150.85 


+ 4.6 


104.6 


150.85 






-j- 2° 12' 


2.33 


233.65 


+ 8.9 


108.9 


233.65 






+ 2° 56' 


3.06 


306.20 


+ 15.7 


115.7 


306.20 






+ 2° 57' 


3.90 


389.99 


+ 20.0 


120.0 


389.99 


BH 


90° 






0.00 




120.0 


T. P. 






+ 0° 30' 


0.67 


68.00 


+ 0.6 


120.6 


457.99 






+ 0° 49' 


1.20 


120.99 


+ 1.7 


121.7 


510.98 






- 0° 28' 


1.84 


185.00 


- 1.5 


118.5 


574.99 


Bi 


116° 52' 






0.00 




118.5 


T. P. 






- 0° 40' 


0.98 


98.99 


- 1.1 


117.4 


673.98 






- 1° 45' 


2.87 


287.74 


- 8.8 


109.7 


862.73 






-0° 2' 


4.96 


497.00 


- 0.3 


118.2 


1071.99 


B.M. 




-0° 8' 


7.30 




- 1.7 


116.8 


B.M. 



IV. PUBLIC LANDS OF THE UNITED STATES 

204. We cannot in this brief treatise enter into the details of the 
survey of the public lands of the United States. The short dis- 
cussion which we give is taken mainly from the " Manual of Sur- 
veying Instructions for the Survey of the Public Lands of the 
United States and Private Land Claims," f to which volume the 
student is referred. Brief accounts of these Government Surveys 
will be found in the works on surveying mentioned. 






* Here B Q, used as a B. M. or T. P., is 100' above datum plane. 

Azimuths are taken with reference to FG, a line established on the previous day. 

■t Published under the direction of the General Land Office, Washington, D.C. 



130 PLANE SURVEYING [Art. 205 

205. The present system of survey of the public lands was in- 
augurated by a committee appointed by the Continental Congress, of 
which Thomas Jefferson was the chairman. The first public surveys 
were made under an ordinance passed in 1785, which provided for 
townships 6 mi. square, containing 36 sections of 1 mi. square, while 
a later Act of Congress directed that the sections be divided into 
quarter sections. 

These " rectangular " divisions were referred to certain well-estab- 
lished lines, — the one a true meridian, the other an east and west 
line, called the base line. 

206. The present law requires that in general the public lands 
of the United States " shall be divided by north and south lines run 
according to the true meridian, and by others crossing them at right 
angles so as to form townships 6 mi. square," and that the corners 
of the townships thus surveyed " must be marked with progressive 
numbers from the beginning." It also provides that the townships 
shall be subdivided into 36 sections, each of which shall contain 640 
acres, as nearly as may be, by a system of two sets of parallel lines, 
one governed by true meridians, and the other by parallels of latitude, 
the latter intersecting the former at intervals of a mile. 

These 36 sections are numbered, commencing with number one at 
the northeast angle of the township, and proceeding west to number 
6, thence proceeding east to number 12, and so on, alternately, to 
number 36 in the southeast angle. 

207. Owing to the convergence of meridians, of course the town- 
ships could not be exactly 6 mi. square, but would be of a trape- 
zoidal form. To prevent this error from accumulating, standard 
parallels are established every 24 mi. north and south of the base 
line, and auxiliary meridians at intervals of every 24 mi. east and 
west of the principal meridian, thus confining the errors resulting 
from convergence of meridians and inaccuracies of measurement 
within comparatively small areas. 

The above is a general outline of the excellent system adopted 
by our government. Partly for the sake of emphasizing what has 
already been said in this book in regard to accuracy in chaining, etc., 
the following articles contain a few of the many instructions given 
to the United States surveyors. 

208. Instruments. — The surveys of the public lands of the United 
States, embracing the establishment of base lines, principal meridians, 



Art. 210] TRANSIT SURVEYING 131 

standard parallels, meander lines, and the subdivisions of townships, 
will be made with instruments provided with the accessories neces- 
sary to deteimine a direction with reference to the true meridian, 
independently of the magnetic needle. 

Burt's improved solar compass, or a transit of improved construc- 
tion, with or without solar attachment, will be used in all cases. 
When a transit without solar attachment is employed, Polaris obser- 
vations and the retracements necessary to execute the work in accord- 
ance with existing law and the requirements of these instructions 
will be insisted upon. 

Deputies using instruments with solar apparatus will be required 
to make observations on the star Polaris at the beginning of every 
survey, and whenever necessary to test the accuracy of the solar appa- 
ratus. 

The township and subdivision lines will usually be measured by 
a two-pole chain 33 ft. in length, consisting of 50 links (each 
= 7.92 in.). On uniform and level ground, however, the four- 
pole chain may be used. The measurements will, however, always 
be expressed in terms of the four-pole chain of 100 links. The 
chain in use must be compared and adjusted with a standard chain 
each working day. 

209. Leveling the Chain and plumbing the Pins. — The length of every 
surveyed line will be ascertained by precise horizontal measurement, 
as nearly approximating to an air-line as is possible in practice on the 
earth's surface. This all-important object can be attained only by a 
rigid adherence to the three following observances : 

First. — Ever keeping the chain drawn to its utmost degree of 
tension on even ground. 

Second. — On uneven ground, keeping the chain not only 
stretched as aforesaid, but leveled. And when ascending and de- 
scending steep ground, hills, or mountains, the chain will have to be 
shortened to one-half or one-fourth its length (and sometimes more), 
in order accurately to obtain the true horizontal measure. 

Third. — The careful plumbing of the tally pins, so as to attain 
precisely the spot where they should be stuck. The more uneven the 
surface, the greater the caution needed to set the pins. 

210. Marking Lines. — All lines on which the legal corner boun- 
daries are to be established will be marked after this method ; viz., 
those trees which may be intersected by the line will have two chops 



132 PLANE SURVEYING [Art. 210 

or notches cut on the sides facing the line, without any other marks 
whatever. These are called "sight trees" or "line trees." A sufficient 
number of other trees standing within 50 links of the line, on either 
side of it, will be blazed on two sides diagonally or quartering toward 
the line, in order to render the line conspicuous, and readily to be 
traced, the blazes to be opposite each other, coinciding in direction 
with the line where the trees stand very near to it, and to approach 
nearer each other toward the line, the farther the line passes from 
the blazed trees. 

These instructions state, moreover, that the required blazes will 
be made on any tree not smaller than 2 in. in diameter, and that 
bushes on or near the line should be bent at right angles with the 
line at about the usual height of blazes. Many other details are 
specified in regard to the careful execution of the work and its 
permanence, especially with respect to the establishment of durable 
monuments to mark the corners. 



CHAPTER VI 



LEVELING 



211. Leveling is the art of determining the relative position of 
points from the centre of the earth. A line is on a true level, when 
every point of it is equally distant from the centre of the earth, and 
a surface is a true level surface when all points on it are equally distant 
from the centre of the earth. A straight line tangent to a line of 
true level is sometimes called a line of apparent level. It is this latter 
line that is determined by the line of collimation of the telescope of 
a leveling instrument. 



212. Thus (Fig. 70) if O is the centre of 
the earth, and ABE a line of true level, AB 
is the line of apparent level. The difference 
between the apparent and true level of the 
points A and B, is BB. This difference, 
due to the curvature of the earth, is com- 
puted as follows : 




Fia. 70 



From geometry, we have 

AB 2, = BB(BB + 2 BO). 

Now for practical purposes, AB does not differ sensibly from AB, 
and BB is so small in comparison with 2 BO that it may be neglected, 
and the formula becomes 



AB 2 = BBx 2 BO, 



or 



BB = 



AB Z 
2 BO 



(1) 



that is, the correction for curvature is equal to the square of the distance 
divided by the diameter of the earth. 

From Formula (1), the curvature in inches or feet may be com- 
puted for any distance. Tables of curvature are thus computed. 

133 



134 PLANE SURVEYING 



[Art. 212 



If AD= 1 mi., BD= 8.001 in., or two-thirds of a foot, very nearly ; 
and for any other distance d, in miles, we have 

l 2 : d 2 = | of a foot : x feet. 

.*. x, in feet, = J d 2 ; 

that is, the following rule gives, approximately, the curvature in feet : 
The correction for curvature, in feet, is equal to tivo-thirds of the 
square of the distance in miles. 

Refraction acts in a direction opposite to curvature, tending to 
lessen the effect of the latter. 

The instruments used in leveling are described in Chapter I. 

213. We shall consider leveling under three heads : 

First. — Differential leveling, which consists in determining the 
difference of level between two given points. 

Second. — Profile leveling, which is the operation of obtaining a 
section or profile along a given line, as a railroad for example. 

Third. — Topographical leveling, which is equivalent to getting 
the profiles of many different lines, for the purpose of obtaining the 
elevations and depressions of the ground over a more or less extended 
area. 

DIFFERENTIAL LEVELING 

214. To determine the difference of level between two points, A 
and B, visible from each other, set up the level * at a point about the 
same distance from A and B, but not necessarily on a line between 
them, and having leveled the instrument, sight a rod held on A and 
then on B ; the difference of the two readings is the difference of 
elevation of A and B. After turning the telescope on A, and again 
after turning it on B, see that the bubble remains in the centre of its 
tube. This point must not be neglected in using any level. For 
short distances the corrections for refraction and curvature are small, 
and are often disregarded. The effect of both, however, is counter- 
acted by setting up the instrument at a point equally distant from 
the two points sighted. Let the student prove this by means of a 
diagram. 

One setting of the level is usually sufficient, provided that the 
two points are visible from the point where the level is set up, and 

* A Y-level, and any good level, or a transit with telescope-bubble, may be used. 



Art. 215] 



LEVELING 



135 



are not much over 1000 ft. apart, and their difference of elevation is 
not greater than 14 or 15 ft. In other cases we proceed as follows : 

215. Suppose the difference of elevation between any two points, 
A and H in Fig. 71, is required. 

Set up the instrument at some point equally distant from the 
initial point A and some other point, B, where it will be convenient 
to hold the rod. No appreciable error will be made if the instru- 
ment is not exactly equally distant from A and B, it being sufficient 
for the surveyor to judge the distances by the eye, or to let the rod- 
man step off the distances. If haste rather than accuracy is desired, 




Fig. 71 



no effort need be made to put the instrument midway between the 
points where the rod is held. Where very great precision is desired, 
a correction for both curvature and refraction should be applied. No 
effort is made to hold the rod on the line between A and H, and the 
instrument is seldom exactly on the line between A and B, or B and 
C, etc., but unnecessary deviation from a direct line is to be avoided. 
Now get the line of sight on the rod at A and take the reading, in 
this case 9.5 (feet), which is usually termed a back-sight, but which 
we shall call a plus sight ; then revolve the telescope and read the 
rod held at B, 2.6, which is usually called a fore-sight, but which we 
shall call a minus sight,* in accordance with the following definitions: 

A plus sight, or reading, is any reading taken on a point of known 
or assumed elevation for the purpose of determining the elevation 
of the instrument (that is, of the line of sight). 

A minus sight, or reading, is any reading- taken for the purpose of 
determining the elevation of the (unknown) point on which the rod 
is held. 

Having noted these + and 



sights in their proper column, as 



* The terms "back-sight" and "fore-sight," while still used by many surveyors, 
are misleading. The author has often seen a beginner puzzled over getting a "fore- 
sight" when the telescope is pointing back along the line, and vice versa. 



136 



PLANE SURVEYING 



[Art. 215 



shown in the form for field notes given below, move the instrument 
to a position, " 2," in advance of B, take the reading (+7.3) of 
the rod held at B, and then the reading (—1.1) of the rod at C. 



Form of Record for Differential Leveling 









B.M. 




No. of 
Sta. 


+ S. 


-s. 




Kemarks 












+ 






1 


9.5 


2.6 


8.24 




B. M. near A is 100 ft. 


2 


7.3 


1.1 






above datum plane. 


3 


4.0 


6.2 








4 


2.3 


9.7 








5 


9.0 


0.7 








6 


9.5 


2.1 








7 


7.1 


2.6 




2.84 


on B. M. near H. 


+ 48.7 


-25.0 




-25.0 










+ 23.7 



Move the instrument to position number " 3 " and proceed as be- 
fore, and so on till the last (minus) sight is taken on the rod held 
at H. 

Then the algebraic sum of all the readings equals the difference 
of elevation of A and H. In this case the difference of elevation 
= 4-23.7, which shows that the last point, H, is higher than A. 

Should the difference of elevation of any intermediate points, as 
B and U, be desired, we simply take the algebraic sum of the -f and 

— sights in the 2d, 3d, and 4th rows above, giving + 13.6 — 17.0 =? 

— 3.4, the minus sign showing that E is below B. 

216. Instruments. — To obtain the best results the instrument 
used should be adjusted every day, and the turning-points should be 
on firm ground. The rodman must hold the rod vertically. The 
best way to do this is to stand behind the rod and hold it loosely 
between the hands, so that it will balance itself.* This is easily 
done if the wind is not high. If a target rod is used, the rodman 
sets the target, the observer directing him by a motion of the hand, 
up or down, till the zero of the target is on the line of sight, when 
the rodman records the reading in his note-book. 

* In precise work, rod-levels are used which, by the position of the bubble, show 
when the rod is vertical. 



Art. 218J LEVELING 137 

If a self-reading rod is used, the observer reads the rod, and 
records the reading in his note-book. Greater speed can be made 
with a self-reading rod. 

There are many different forms of levels, some of which are de- 
scribed in Chapter I, where will also be found a description of two 
leveling-rods (Art. 96). 

217. Bench-marks and Turning-points. — Bench-marks are fixed 
points whose elevations are known or assumed to be known. Turn- 
ing-points are temporary bench-marks, and serve as reference points 
when the instrument is moved and set up elsewhere. All the inter- 
mediate points, B, C, D, etc., in the last example, are really turning- 
points. Their position (as to elevation) is fixed by the minus (or 
fore) sight taken before the instrument is moved. Then, after the 
instrument is moved, the next point is determined with reference to 
this turning-point, and so on. 

A bench-mark (B. M.) is denoted in various ways, such as a stake 
driven in the ground, a spike in the root of a tree, a stone pillar, or 
a point on a natural rock. 

A turning-point (T. P.) is usually something less durable, as a 
small peg, or even the hard crust of a roadbed. 

In the field notes, usually under the head of remarks, the exact 
position of all bench-marks should be described. In important work 
the bench-mark should be of so permanent a character and so well 
described that a surveyor could find it many years later. 

If getting the difference of elevation between two points occupies 
only a few hours, bench-marks may be unnecessary, but, if the work 
is interrupted, as for dinner or at nightfall, a bench-mark must be 
established and its position recorded in the note-book. 

PROFILE LEVELING 

218. In profile leveling the object is to get the profile of certain 
lines established on the ground. 

Such a profile is needed in many kinds of work, such as obtaining 
the grade of a railroad, highway, water-pipe, or drain-pipe line. 
Profile leveling differs from differential leveling in that the distance 
from some initial point of each position of the rod, which is always 
held on certain lines, is recorded together with the elevation of the 
point above a certain datum plane. 

This datum plane is usually taken a certain number of feet below 
a well defined bench-mark, and, to avoid minus signs, it is best to 



138 



PLANE SURVEYING 



[Art. 21S 



assume it lower than the lowest point of the profile,* preferably an 
even hundred of feet below the bench-mark. 

219. In profile leveling the height of the instrument is first ob- 
tained by taking a reading on a rod held upon a bench-mark, the 
height of this B. M. above the plane assumed as a datum plane being 
known. This reading upon the B. M. is a plus sight, and the height 
of the instrument (H. I.) is this reading added to the elevation of 
the B. M. 

The rod is now held at some point on the line ; if at the beginning 
this may be called station " 0." This is a minus sight, and is recorded 
on the second line, opposite station " 0." We give below a form of re- 
duction, which differs but little from a, sample page given in Johnson's 

Form of Record for Profile Leveling 



+ s. 


B.M. 

OR 

T. P. 


H.I. 


- s. 


S. E. 


Stations 


Remarks 


2.810 


100.00 


102.810 


5.81 

8.61 
9.94 


97.00 
94.20 

92.87 


B.M. (a) 
O 
1 
2 




1.620 


92.315 


93.935 


10.495 
2.96 
1.80 
4.90- 
5.27 


90.97 
92.13 

/ 89.03 
88.66 


B.M. (6) 

3 

3 + 40 

4 

6 




1.481 


89.657 


91.138 


4.278 

4.62 

3.12 


86.51 

88.01 


B.M. (c) 
7 
8 




3.355 


88.658 
86.540 


92.013 


2.48 
2.07 
4.20 
6.20 
5.473 


89.94 
87.81 
85.81 


T. P. 

9 
10 
11 

B. M. (d) 





In the above form + S. denotes plus sight; — S., minus sight; B.M., bench- 
mark; T.P., turning-point; H.I., height of instrument; S.E., surface elevation. 

"Surveying." f This form requires six columns, all of which can be 
put on the left-hand page of the note-book, leaving the right-hand 
page for remarks. The 1st, 4th, and 6th columns are taken in the 

* In tide-water country, the elevation of mean tide is often assumed as the datum 
plane. 

t A contribution to the Engineering News by Mr. E. S. Walters, a railroad engineer. 



Art. 221] LEVELING 139 

field ; the 2d, 3d, and 5th can be filled in afterwards. The calculation 
is so simple that the surveyor can usually find time to complete these 
in the field, and thus often has an opportunity to detect errors on 
the spot. Under the head of " Remarks," the surveyor writes out 
full information in regard to bench-marks, etc. In this example, the 
first and last bench-marks are thus described : 

B. M. (a), southwest corner lower step, front door Library, cross (X) in stone. 
B. M. (d), an iron spike in stump, 2' from sidewalk in Elliott Park, nearly opposite 
Miss Elliott's cottage. 

220. The work moves along more rapidly if the line is measured 
out previously and stakes are set at distances of 100 ft., or wherever 
it is deemed expedient to take the elevation. 

Sometimes, generally on account of an abrupt change in the 
ground, it is desirable to take readings at shorter distances than 
100 ft. ; the distance is then called a plus. In our notes there is 
such a point at the distance of 340 ft., which is recorded as station 
"3 + 40." In like manner " 26 + 30 " would denote a station 2630 ft. 
from the starting-point. 

If at any point the elevation changes very little, the rod need not 
be held at every 100 ft. mark. In our example, station " 5 " is not 
given, and an examination of the S. E. column shows that there is 
but little change between 400 and 600 ft. In this instance, a B. M. 
was usually made instead of a T. P. when the instrument was moved, 
because these points were at the intersection of side streets, and 
bench-marks were left for reference in running profiles of these 
streets. The H. I. values are necessary to deduce the S. E. values, 
and must be recorded ; but the elevations of turning-points, being 
steps in the calculation of the H. I., need not necessarily be recorded. 

The elevations given in column 2 are not inserted in column 
5 because they form no part of the surface elevations that are 
sought. Occasionally a T. P. may coincide with a station, in which 
case the — S. of the T. P. is the same as the — S. of a station, and the 
elevation of that T. P. is the same as the S. E. of the station; but, 
to avoid confusion, it is best to record the readings in separate lines. 

221. To get the H. I. in first position, we add the +S. to the 
elevation of the B. M. 

To get the H. I. in second position,* add to the elevation of 

* Caution. — Before moving the instrument, be sure to take a sight (a minus sight) 
on the next T. P. or B. M. After the instrument is set up, the first sight taken is tlie 
plus sight on this T. P. or B. M. 



140 



PLANE SURVEYING 



[Art. 221 



B. M. (6) the +- S.; or, regardless of sign, subtract from the first H. I. 
the — S. and add to the result the + S. of the B. M. (that is, add 
algebraically to the first H. I. the -S. and +S. of the T. P. or B. M.); 

thus ' 102.810 - 10.495 + 1.620 = 93.935. 

For the H. I. in third position, 

93.935 - 4.278 + 1.481 = 91.138, 
and so on. ; 

The surface elevations are obtained by subtracting numerically 
(or adding algebraically) the minus sights from the last H.I. found; 

thus S. E. at = 102.81 - 5.81 = 97.00 

S. E. at 1 = 102.81-8.61 = 94.20 

S. E. at 2 = 102.81-9.94 = 92.87 

S. E. at 3= 93.93-2.96 = 90.97 

etc. 

Readings on bench-marks or turning-points are taken to thou- 
sandths, other readings to hundredths, and the surface elevations 
are computed to hundredths, or often only to tenths. 

In the field notes below, fill out properly the 2d, 3d, and 5th 
columns. The field practice of the student will furnish many simi- 
lar examples. . 

Example 



+ s. 


B.M. orT.P. 


H.I. 


-s. 


S.E. 


Stations 


0.673 


50.000 


50.673 


1.00 
0.90 
5.60 
12.80 
6.62 




B.M. 
O 

1 
2 

2 + 90 

3 + 50 


11.482 






5.631 

8.80 




T.P. 
5 


12.043 






1.632 
10.23 
5.00 
3.30 




T.P. 
6 
7 
8 


11.692 




(78.071) 


0.556 

8.52 

6.20 




T.P. 

9 
10 



Art. 223] 



LEVELING 



141 



ESTABLISHING A GRADE LINE 

222. The final grade depends upon the character of the ground 
and the purpose for which the line is run. A grade that will answer 
for a highway is often much too steep for a railroad. It might be 
expedient to spend thousands of dollars to reduce the grade of a rail- 
road while it would be folly to spend a hundred dollars to lessen a 
similar grade on a highway. It is beyond the scope of this work to 
discuss the various conditions that should have weight in deciding 
upon the grade to be adopted. 

One example is sufficient to give an idea of the process of estab- 
lishing a grade line after the surface elevations have been obtained, 
as in Art. 219. The following is a suitable reduction form:* 



Stations 


S.E. 


Grade 


Cut 
+ 


Fill 


Remarks 





41.0 


41.0 


0.0 


0.0 




1 


40.5 


38.7 


1.8 






2 
3 


40.5 
33.6 


36.4 
34.1 


4.1 


0.5 


at 2.96 


3 + 40 
4 


29.5 
32.6 


33.0 
31.6 


1.0 


3.5 


at 3.90 


5 


35.2 


29.4 


5.8 






6 
7 


32.0 
21.0 


27.0 
24.6 


5.0 


3.6 


at 6.64 


8 


16.0 


22.3 




6.3 




9 


19.8 


19.8 


0.0 


0.0 





223. The Grade Line. — Figure 72 is the profile of 
Suppose the grade must not be steeper than 2.5 in 100. 
ence of elevation between 
" 0" and "9" is 21.2 ft. 
Now 21.2 in 900 is 2.36 
in 100, which is within 
our limit. Moreover, by 
drawing a line from "0" 
to " 9," we observe that 
the excavation and em- 
bankment are not very 
different, the cuts being 



the above. 
The difler- 




4 5 
Fig. 72 



a little in excess of the fills ; this 



is 



* This is evidently not from the profile of Art. 219. These surface elevations were 
deduced from a similar profile. 



142 PLANE SURVEYING [Art. 223 

usually an advantage, as most earth shrinks when excavated and 
removed. 

Having determined on the grade line, the column of grade 
heights (column 3) is easily filled out either by allowing a descent 
of 2.36 ft. from the "0" point for each 100 ft., or by taking the 
grade numerals at once from the diagram (Fig. 72) by means of 
the proper scale. 

If coordinate paper is used, this graphic method is very rapid and 
in most cases sufficiently accurate. 

It is customary to use a larger scale for the vertical than for the 
horizontal distances, thus magnifying the inequalities of the profile. 
If this were not done, only very large inequalities would show on the 
diagram. The scale used in Fig. 72 is a very suitable one ; viz. for 
the horizontal, 1 in. to 100 ft., and for the vertical, 1 in. to 10 ft. 

It is often economical and advantageous, for other reasons, to 
change the grade even within the short space of a few hundred feet. 

TOPOGRAPHICAL LEVELING 

224. In topographical surveying it is necessary to determine the 
distances of points above (or below) a certain datum plane, as well 
as to find their positions with reference to certain fixed lines, such as 
a meridian and a parallel of latitude, the latter measurements being 
made in a horizontal plane. 

The objects of such a survey are many. A topographical survey 
over the proposed route of a railroad should be made in order that a 
safe, economic road be constructed. Whenever a town or city is to 
be laid out, a topographical survey is absolutely necessary if the best 
results are to be obtained in locating streets and selecting the routes 
for water and sewer pipes. 

When a ravine is to be converted into a reservoir by a dam thrown 
across it, a topographical survey furnishes the means of telling to 
what distance the water will back up the ravine and of calculating 
the contents of the reservoir. 

Such a survey will likewise furnish the data for computing the 
amount of earth to be excavated in making a reservoir, leveling a 
field for an athletic ground, or for any other purpose. 

225. Topographical leveling consists essentially in getting a series 
of profiles of the ground to be surveyed. There are several methods 
arising from the different arrangements of this series of profiles and 
from the instruments used. 



Art, 226] 



LEVELING 



143 



The old method of getting the elevations of each point, derived 
by means of a level and measuring the horizontal distances with a 
chain, or tape, and getting azimuths or bearings with a transit or 
compass, is accurate, but very laborious and costly, and is chiefly 
employed when the area of the survey is very limited and great 
accuracy is required. 

The plane-table, either with or without stadia wires, has been 
very extensively used for this purpose. Both of these methods are 
being superseded by the stadia method, in which the transit and 
stadia rods are the only instruments used. The azimuths, distances, 
and elevations are quickly obtained. (See Chapter V, iii.) 

If the work is based on a rigid triangulation system, the accuracy 
of this method is usually all that is desired. We cannot enter into 
any description of the field work, but what has already been said 
under the heads of transit surveying, stadia work, and profile level- 
ing will serve as a guide to the student or surveyor who undertakes 
such work. 



1 2 


2 2 




2 1 


4 


4 


4 


4 


4 


4 


4 


4 


4 


4 


4 


4 


4 


3 
2 




2 1 











226. Grading. — A very simple method of grading, or finding the 
amount of excavation or embankment, is as follows : 

Divide the area into rectangles (Fig. 73), the longer sides of 
which should not be over 50 ft. on ordinary rolling ground, and 
drive pegs at the corners. Find 
the elevation at each intersec- 
tion by means of a level, referring 
the elevations to some datum 
plane below the surface after it 
is graded. Subtract from these 
readings the elevations of the cor- 
responding points on the graded 
surface. The several differences 
are the depths of excavation (or fill) at the corners. The contents of 
any partial volume is the mean of the four corner heights multiplied 
by the area of its cross-section. Now, since the rectangular areas 
were made equal, and since each corner height will be used as many 
times as there are rectangles joining at that corner, we have, in 
cubic yards, 

V= ^^7 & h i + 2 27 '2 + 3 ^h + 4 2AJ, 

where V— the contents sought, A = area of each cross-section, 2/^ 
== the sum of all the corner heights that stand alone, 2A 2 = the sum 



Fig. 73 



144 PLANE SURVEYING [Art. 226 

of all the corner heights common to two rectangles, etc., the subscripts 
denoting the number of adjoining rectangles. 
Hence we have the 

Rule. — Take each corner height as many times as there are 
partial areas adjoining it, add them all together, and multiply by one- 
fourth of the area of a single rectangle. This gives the volume in 
cubic feet. To obtain it in cubic yards, divide by twenty-seven. 

The calculation is simplified if the ground be laid out in rectangles 
30 ft. by 36 ft., for then 

A 1080 



4 x 27 108 



= 10. 



TABLES 



TABLES 

The first ten tables will be found in the body of the book, as 
follows : 

PAGE 

Table I. — Mean Refraction 36 

II. — Errors in Azimuth ......... 37 

III. — Refraction Correction, Lat. 40° 39 

IV. — Latitude Coefficients .42 

V. — Daily Variation of the Needle 75 

VI. — Declination Formulae 76 

VII. — Declination Values and Annual Change . . . . .78 

VIII. — Azimuths of Polaris at Elongation 81 

IX. — Local Mean Time of Culminations and Elongations of Polaris 82 
X. — Pole Distance of Polaris ........ 88. 



TABLE XI 

(1) Linear Measure 

12 inches (in.) make 1 foot . . 

3 ft. make 1 yard . . 

5| yd. make 1 rod or pole 

40 rd. make 1 furlong 

8 fur. make 1 mile . . 



ft. 

yd. 

rd. or p. 

fur. 

mi. 



Equivalents 



mi. fur. rd. yd. ft. 

1 = 8 = 320 = 1760 = 5280 
1 = 40 = 220 = 660 



(2) Surveyors' Linear Measure 



7.92 inches make 1 link . . 

25 1. make 1 rod or pole 

4 rd., or 66 ft., make 1 chain . . 

80 ch. make 1 mile . . 

Equivalents 
mi. ch. rd. ft. 

1 = 80 = 320 = 5280 
1 = 4 = 66 
1 = 16J 
147 



1. 

rd. or p. 

ch. 



148 



METRIC SYSTEM 



(3) Square Measure 

144 square inches (sq. in.) make 1 square foot 

9 sq. ft. make 1 square yard 

30^ sq. yd. make 1 square rod 

40 sq. rd. make 1 rood . . 

4 R. make 1 acre . . . 

640 A. make 1 square mile 

Equivalents 

A. R. sq. rd. sq. yd. sq. ft. 

1 = 4 = 160 = 4840 = 43560 
1 = 40 = 1210 = 10890 
1 = 30i = 272J 



sq. ft. 
sq. yd. 
sq. rd. 
R. 
A. 
sq. mi. 



(4) Surveyors' Square Measure 

625 square links make 1 pole (sq. rd.) 
16 P. make 1 square chain 

10 sq. ch. make 1 acre . . 

640 A. make 1 square mile . 

36 sq. mi. make 1 township 



P. 

sq. ch. 
A. 

sq. mi. 
Tp. 



(5) Cubic Measure 

1728 cubic inches (cu. in.) make 1 cubic foot 
27 cu. ft. make 1 cubic yard 

16 cu. ft. 
8 cd. ft. or 
128 cu. ft. 
24f * cu. ft. make 1 perch of stone 



make 1 cord foot 
make 1 cord of wood 



cu. ft. 
cu. yd., 
cd. ft. 

cd. 

Pch. 



METRIC SYSTEM 



(6) Linear 

10 millimetres = 

10 centimetres = 

10 decimetres = 

10 metres = 

10 dekametres = 

10 hektometres = 
10 kilometres 



Measure 

= 1 centimetre 

= 1 decimetre 
- 1 metre 
= 1 dekametre 
= 1 hektometi*3 
= 1 kilometre 
= 1 myriametre. 



(7) Square Measure 



100 square 

100 square 
100 square 
100 square 
100 square 
100 square 



millimetres 

centimetres 

decimetres 

metres 

dekametres 

hektometres 



1 square centimetre 
1 square decimetre 
1 square metre 
1 square dekametre 
1 square hektometre 
1 square kilometre 



* Varies in different localities. 



METRIC SYSTEM 



149 



Also, A centare = a square metre 

An are = a square dekametre, or 100 centares 
A hektare = a square hektometre, or 100 ares 

Note. — The square metre is used in measuring ordinary surfaces; the square 
kilometre, in measuring the areas of countries ; the are and hektare, in measuring land. 

(8) Cubic Measure 

1000 cubic millimetres = 1 cubic centimetre 
1000 cubic centimetres = 1 cubic decimetre 
1000 cubic decimetres = 1 cubic metre 

The following terms are also used : 

A decistere = .1 cubic metre 

A stere = 1 cubic metre, or 10 decisteres 

Note. — The cubic metre is used in measuring ordinary solids; the stere, in 
measuring wood. 



(9) For Converting Metres* Feet, and Chains 



Metres to Feet 


Feet to Metres and 


Chains 


Chains to Feet 


Metres 


Feet 


Feet 


Metres 


Chains 


Chains 


Feet 


1 


3.28087 


1 


0.304797 


0.0151 


0.01 


0.66 


2 


6.56174 


2 


0.609595 


.0303 


.02 


1.32 


3 


9.84261 


3 


0.914392 


.0455 


.03 


1.98 


4 


13.12348 


4 


1.219189 


.0606 


.04 


2.64 


5 


16.40435 


5 


1.523986 


.0758 


.05 


3.30 


6 


19.68522 


6 


1.828784 


.0909 


.06 


3.96 


7 


22.96609 


7 


2.133581 


.1061 


.07 


4.62 


8 


26.24695 


8 


2.438378 


.1212 


.08 


5.28 


9 


29.52782 


9 


2.743175 


.1364 


.09 


5.94 


10 


32.80869 


10 


3.047973 


.1515 


.10 


6.60 


20 


65.61739 


20 


6.095946 


.3030 


.20 


13.20 


30 


98.42609 


30 


9.143918 


.4545 


.30 


19.80 


40 


131.2348 


40 


12.19189 


.6061 


.40 


26.40 


50 


164.0435 


50 


15.23986 


.7576 


.50 


33.00 


60 


196.8522 


60 


18.28784 


.9091 


.60 


39.60 


70 


229.6609 


70 


21.33581 


1.0606 


.70 


46.20 


80 


262.4695 


80 


24.38378 


1.2121 


.80 


52.80 


90 


295.2782 


90 


27.43175 


1.3636 


.90 


59.40 


100 


328.0869 


100 


30.47973 


1.5151 


1 


66.00 


200 


656.1739 


200 


60.95946 


3.0303 


2 


132 


300 


984.2609 


300 


91.43918 


4.5455 


3 


198 


400 


1312.348 


400 


121.9189 


6.0606 


4 


264 


500 


1640.435 


500 


152.3986 


7.5756 


5 


330 


600 


1968.522 


600 


182.8784 


9.0909 


6 


396 


700 


2296.609 


700 


213.3581 


10.606 


7 


462 


800 


2624.695 


800 


243.8378 


12.121 


8 


528 


900 


2952.782 


900 


274.3175 


13.636 


9 


594 


1000 


3280.869 


1000 


304.7973 


15.151 


10 


660 


2000 


6561.739 


2000 


609.5946 


30.303 


20 


1320 


3000 


9842.609 


3000 


914.3918 


45.455 


30 


1980 


4000 


13123.48 


4000 


1219.189 


60.606 


40 


2640 


5000 


16404.35 


5000 


1523.986 


75.756 


50 


3300 


6000 


19685.22 


6000 


1828.784 


90.909 


60 


3960 


7000 


22966.09 


7000 


2133.581 


106.06 


70 


4620 


8000 


26246.95 


8000 


2438.378 


121.21 


80 


5280 


9000 


29527.82 


9000 


2743.175 


136.36 


90 


5940 



150 MENSURATION FORMULA 

\ 



TABLE XII 
Mensuration Formulae 

In these formulae, S = area ; h = altitude ; a, 5, e, sides of a tri- 
angle ; r = radius of circle. 

(1) Triangle, base and altitude given, S=^bh. 

(2) Triangle, the three sides given, 

s = i i a + & + <?), 



$ = Vs(s — a) (s — 6) (s — <?) . 

(3) Trapezoid, a and 5 being the parallel sides, S= \ (a -f- 5)^. 

(4) Parallelogram, # = M. 

(5) Circle, tf= Trr 2 (tt = 3.1416). 

In the following, S= surface; V= volume; H= altitude; L= slant 
height; R = radius of base; 5 = area of base; P = perimeter of base, 

(6) For prism, S=PH, V= B x H. 

(7) For pyramid, S=± PL, V=\BxH. 

(8) For cylinder, S=2ttRH, V=ttB 2 H. 

(9) For cone, S=irRL, V=\irR 2 H. 
(10) For sphere, S=4ttR 2 , V=±ttR\ 



TRIGONOMETRIC FORMULA 



151 



TABLE XIII 
Trigonometric Formulae 

In the formulse below, a, b, c denote the sides of the triangle 
ABC opposite the angles A, B, C, respectively; s = 1 (a + 6 -|- c), 
and #=area of the triangle. 

(i) To solve a right-angled plane triangle ((7=90°), use one 
or more of the following formulae : 

sin A = -, cos A = -, tan A == -, cot A = -, 
c c o a 

sin 2?=-, cos B=-, tanJ9 = -, coiB=^, 
c c a b 

C 2 =a * + b 2 , A = 90°-B, B = 90°-A. 

(ii) Oblique triangles may be solved by some of the following 
formulas : 





Given 


Sought 


Formulae 


(1) 

(2) 
(3) 

(4) 


A,B, a 

A, a, b 
C,a,b 

a, b, c 


C,b,c 

B,C,c 
A,B,c 

A, etc. 


C=180° (A+B), b= a sin B, 
sin A 

c= a sin (A + B). 
sin A 

sin B = sin A b, C = 180° (.4 + B), c = a sm f , 
a smA 

i (A + B)=90°-i C, 

tan J (.4 B) = ° ~ b tan $ (A + B), 

a + b 

A=%(A + B) + h(A-B), 

B = \ {A + B)-$(A- B), 

, , 7 x cosiM + jB) 

c — (a + b) 2-^— -f-i 

K cos £ (.4 - 5) 

S = % ab sin C. 




ta n M = V ( '"/ )C '7 C)| 
' s(s — a) 

5=V.s(s- «)0 -6)(s- c )- 



152 



SQUARES, CUBES, AND ROOTS 



TABLE XIV 

Squares, Cubes, Square Roots, and Cube Roots 



No. 


Square 


Cube 


Sq. Root 


Ou, Root 


No. 


Square 


Cube 


Sq. Root 


Cu. Root i 


i 


! 


1 


1. 0000 


1. 0000 


5i 


2601 


132651 


7.1414 


3.7084 


2 


. 4 


8 


1.4142 


1-2599 


52 


2704 


140608 


7.21 1 1 


37325 


3 


9 


27 


1.7320 


1.4422 


53 


2809 


148877 


7.2801 


37563 


4 


16 


64 


2.0000 


1.5874 


54 


2916 


157464 


7-3485 


37798 


5 


25 


125 


2.2361 


1. 7100 


55 


3025 


166375 


7.4162 


3.8030 


6 


36 


216 


2.4495 


1.8171 


56 


3136 


175616 


7-4833 


3.8259 


7 


49 


343 


2.6458 


1.9129 


57 


3249 


185193 


7.5498 


38485 


8 


64 


512 


2.8284 


2.0000 


58 


3364 


195112 


7.6158 


3.8709 


9 


81 


729 


3.0000 


2.0801 


59 


348i 


205379 


7.681 1 


3.8930 


IO 


100 


1000 


3-J623 


2.1544 


60 


3600 


216000 


7.7460 


3.9149 


ii 


121 


i33i 


3.3166 


2.2240 


61 


3721 


226981 


7.8102 


3.9365 


12 


144 


1728 


3.4641 


2.2894 


62 


3844 


238328 


7.8740 


3.9579 


13 


169 


2197 


3.6056 


2-35*3 


63 


3969 


250047 


7-9373 


3-9791 


14 


196 


2744 


3.7417 


2.4101 


64 


4096 


262144 


8.0000 


4.0000 


15 


225 


3375 


3-8730 


2.4662 


65 


4225 


274625 


8.0623 


4.0207 


16 


256 


4096 


4.0000 


2.5198 


66 


4356 


287496 


8.1240 


4.0412 


17 


289 


49i3 


4-1231 


2.5713 


67 


4489 


300763 


8.1854 


4.0615 


18 


324 


5832 


4.2426 


2.6207 


68 


4624 


3H432 


8.2462 


4.0817 


19 


36i 


6859 


4-3589 


2.6684 


69 


4761 


328509 


8.3066 


4.1016 


20 


400 


8000 


4.4721 


2.7144 


70 


4900 


343000 


8.3666 


4-1213 


21 


441 


9261 


4.5826 


2.7589 


7i 


5041 


3579H 


8.4261 


4.1408 


22 


484 


10648 


4.6904 


2.8020 


72 


5184 


373248 


8.4853 


4.1602 


23 


5 2 9 


12167 


4-7958 


2.8439 


73 


5329 


389017 


8.5440 


4.1793 


24 


576 


13824 


4.8990 


2.8845 


74 


5476 


405224 


8.6023 


4.1983 


25 


625 


15625 


5.0000 


2.9240 


75 


5625 


421875 


8.6603 


4.2172 


26 


676 


17576 


5.0990 


2.9625 


76 


5776 


438976 


8.7178 


4.2358 


27 


729 


19683 


5.1962 


3.0000 


-77 


5929 


456533 


8-7750 


4-2543 


28 


784 


21952 


5-29I5 


3.0366 


78 


6084 


474552 


8.8318 


4.2727 


29 


841 


24389 


5-3852 


3-0723 


79 


6241 


493039 


8.8882 


4.2908 


30 


900 


27000 


5-4772 


3.1072 


80 


6400 


5 1 2000 


8.9443 


4.3089 


31 


961 


29791 


5.5678 


3-HI4 


81 


6561 


53H4I 


9.0000 


4.3267 


32 


1024 


32768 


5-6569 


3.1748 


82 


6724 


551368 


9.0554 


4-3445 


33 


1089 


35937 


5-7446 


3.2075 


83 


6889 


571787 


9. 1 104 


4.3621 


34 


1156 


393°4 


5-8310 


3.2396 


84 


7056 


592704 


9.1652 


4-3795 


35 


1225 


42875 


5-9i6i 


3-2711 


85 


7225 


614125 


9.2195 


4.3968 


36 


1296 


46656 


6.0000 


3-3019 


86 


7396 


636056 


9.2136 


4.4140 


37 


1369 


50653 


6.0828 


3-3322 


87 


7569 


658503 


9-3274 


4.4310 


38 


1444 


54872 


6.1644 


3.3620 


88 


7744 


681472 


9.3808 


4.4480 


39 


1521 


59319 


6.2450 


3-3912 


89 


7921 


704969 


9.4340 


4.4647 


40 


1600 


64000 


6.3246 


3.4200 


90 


8100 


729000 


9.4868 


4.4814 


4i 


1681 


68921 


6.4031 


3.4482 


9i 


8281 


753571 


9-5394 


4-4979 


42 


1764 


74088 


6.4807 


3.4760 


92 


8464 


778688 


9-5917 


4-5 r 44 


43 


1849 


795°7 


6-5574 


3-5°34 


93 


8649 


8o4357 


9.6437 


4.5307 


44 


1936 


85184 


6.6332 


3-5303 


94 


8836 


830584 


9.6954 


4.5468 


45 


2025 


91125 


6.7082 


3-5569 


95 


9025 


857375 


9.7468 


4.5629 


46 


2116 


97336 


6.7823 


3-5830 


96 


9216 


884736 


9.7980 


4.5789 


47 


2209 


103823 


6.8557 


3.6088 


97 


9409 


912673 


9.8489 


4-5947 


48 


2304 


1 10592 


6.9282 


3-6342 


98 


9604 


941 192 


9.8995 


4.6104 


49 


2401 


1 1 7649 


7.0000 


3.6593 


99 


9801 


970299 


9.9499 


4.6261 


50 


2500 


125000 


7.071 1 


3.6840 


100 


1 0000 


1 000000 


10.0000 


4.6416 



CHORDS 



153 



TABLE XV 

Chords* 






0' 


10' 


20' 


30' 


40' 


5o' 


d. 


P.P. 





0.0000 


0.0029 


0.0058 


0.0087 


0.0116 


0.0145 


29 




I 


0.0175 


0.0204 


0.0233 


0.0262 


0.0291 


0.0320 


29 




2 


0.0349 


0.0378 


0.0407 


0.0436 


0.0465 


0.0494 


29 


30 


3 


0.0524 


0.0553 


0.0582 


0.0611 


0.0640 


0.0669 


29 


1 


30 
60 
90 


4 


0.0698 


0.0727 


0.0756 


0.0785 


0.0814 


0.0843 


29 


2 

3 


5 


0.0872 


0.0901 


0.0931 


0.0960 


0.0989 


0.1018 


29 


6 


0.1047 


0.1076 


0.1 105 


0.1 134 


0.1163 


0.1192 


29 


4 


12 


7 


0.1221 


0.1250 


0.1279 


0.1308 


O.I337 


0.1366 


29 


5 


150 


8 


o.i395 


0.1424 


O.I453 


0.1482 


0.1511 


0.1540 


29 


6 


18 


9 


0.1569 


0.1598 


0.1627 


0.1656 


0.1685 


0.1714 


29 


7 
8 


21 
24 


10 


O.I743 


0.1772 


0.1801 


0.1830 


0.1859 


0.1888 


29 


ii 


0.1917 


0.1946 


O.I975 


0.2004 


0.2033 


0.2062 


29 


9 


27 


12 


0.2091 


0.2119 


0.2148 


0.2177 


0.2206 


0.2235 


29 




13 


0.2264 


0.2293 


0.2322 


0.2351 


0.2380 


0.2409 


29 




14 


0.2437 


0.2466 


0.2495 


0.2524 


0.2553 


0.2582 


29 


29 


15 


0.2611 


0.2639 


0.2668 


0.2697 


0.2726 


0.2755 


29 


1 


29 

58 

87 

116 

14 5 
17 4 
20 3 


16 


0.2783 


0.2812 


0.2841 


0.2870 


0.2899 


0.2927 


29 


2 
3 
4 
5 
6 
7 


17 


0.2956 


0.2985 


0.3014 


0.3042 


0.3071 


0.3100 


29 


18 


0.3129 


0.3157 


0.3186 


0.3215 


0.3244 


0.3272 


29 


19 


0.3301 


0.3330 


o.3358 


0.3387 


0.3416 


0-3444 


29 


20 


0-3473 


0.3502 


0.3530 


0.3559 


0.3587 


0.3616 


29 


21 


0.3645 


0.3673 


0.3702 


0.3730 


0-3759 


0.3788 


28 


8 


23 2 


22 


0.3816 


0.3845 


0.3873 


0.3902 


0.3930 


0-3959 


28 


9 


26 1 


23 


0.3987 


0.4016 


0.4044 


0.4073 


0.4101 


0.4130 


28 




24 


0.4158 


0.4187 


0.4215 


0.4244 


0.4272 


0.4300 


28 


28 


25 


0.4329 


o.4357 


0.4386 


0.4414 


0.4442 


0.4471 


28 


26 


0.4499 


0-4527 


0.4556 


0.4584 


0.4612 


0.4641 


28 


1 


28 


27 


0.4669 


0.4697 


0.4725 


0.4754 


0.4782 


0.4810 


28 


2 


56 


28 


0.4838 


0.4867 


0.4895 


0.4923 


0.4951 


o.4979 


28 


3 


84 


29 


0.5008 


0.5036 


0.5064 


0.5092 


0.5120 


0.5148 


28 


4 

5 


11 2 

14 


30 


0.5176 


0.5204 


0.5233 


0.5261 


0.5289 


0.53*7 


28 


31 


0-5345 


0-5373 


0.5401 


0.5429 


0-5457 


o.5485 


28 


6 


16 8 


32 


0.5513 


o.554i 


0.5569 


0.5597 


0.5625 


0.5652 


28 


7 


19 6 


33 


0.5680 


0.5708 


0.5736 


0.5764 


0.5792 


0.5820 


28 


8 


22 4 


34 


0.5847 


0.5875 


0.5903 


0.5931 


0-5959 


0.5986 


28 


9 


25 2 


35 


0.6014 


0.6042 


0.6070 


0.6097 


0.6125 


0.6153 


28 




36 


0.6180 


0.6208 


0.6236 


0.6263 


0.6291 


0.6319 


28 


27 


37 


0.6346 


0.6374 


0.6401 


0.6429 


0.6456 


0.6484 


28 


38 


0.6511 


0.6539 


0.6566 


0.6594 


0.6621 


0.6649 


28 


1 


27 


39 


0.6676 


0.6704 


0.6731 


0.6758 


0.6786 


0.6813 


27 


2 
3 
4 

5 
6 


54 

8 1 

10 8 


40 


0.6840 


0.6868 


0.6895 


0.6922 


0.6950 


0.6977 


27 


4i 


0.7004 


0.7031 


0.7059 


0.7086 


0.7113 


0.7140 


27 


13 5 
16 2 


42 


0.7167 


0.7195 


0.7222 


0.7249 


0.7276 


0.7303 


27 


43 


0.7330 


o.7357 


0.7384 


0.7411 


0.7438 


0.7465 


27 


7 
8 

9 


189 
21 6 
24 3 


44 


0.7492 


o.75 J 9 


0.7546 


0.7573 


0.7600 


0.7627 


27 


45 


0.7654 


0.7681 


0.7707 


0.7734 


0.7761 


0.7788 


27 


46 


0.7815 


0.7841 


0.7868 


0.7895 


0.7922 


0.7948 


27 




47 


o.7975 


0.8002 


0.8028 


0.8055 


0.8082 


0.8108 


27 




48 


0.8135 


0.8161 


0.8188 


0.8241 


0.8241 


0.8267 


26 


26 2& 


49 


0.8294 


0.8320 


0.8347 


0.8373 


0.8400 


0.8426 


26 

26 


1 
2 


26 25 
52 50 


50 


0.8452 


0.8479 


0.8505 


0.8531 


0.8558 


0.8584 


51 


0.8610 


0.8636 


0.8663 


0.8689 


0.8715 


0.8741 


26 


3 


78 75 


52 


0.8767 


0.8794 


0.8820 


0.8846 


0.8872 


0.8898 


26 


4 


10 4 10 


53 


0.8924 


0.8950 


0.8976 


0.9002 


0.9028 


0.9054 


26 


5 


13 12 5 


54 


0.9080 


0.9106 


0.9132 


o.9 J 57 


0.9183 


0.9209 


26 


6 
7 


15 6 15 
18 2 17 s 


55 


0.9235 


0.9261 


0.9287 


0.9312 


o.9338 


0.9364 


26 


56 


0.9389 


0.9415 


0.9441 


0.9466 


0.9492 


0.9518 


26 


8 


20 8 20 


57 


o.9543 


0.9569 


0.9594 


0.9620 


0.9645 


0.9671 


26 


9 


23 4 22 5 


58 


0.9696 


0.9722 


0.9747 


0.9772 


0.9798 


0.9823 


25 




59 


0.9848 


0.9874 


0.9899 


0.9924 


0.9950 


o.9975 


25 






0' 


10' 


20' 


30' 


40' 


50' 


d. 


P.P. 



* Taken from Gauss's Tables. 



154 



CHORDS 



TABLE XV— Concluded 
Chords 



























o 


0' 


10' 


20' 


30' 


40' 


5o' 


d. 


P. 


P. 


60 


1. 0000 


1.0025 


1 .0050 


1.0075 


I.OIOI 


1. 0126 


25 


26 


2 5 


61 


1.0151 


1.0176 


1. 0201 


1.0226 


1.0251 


1.0276 


25 


1 


26 


25 
5 ° 


62 


1. 0301 


1.0326 


1.0351 


1 -0375 


1.0400 


1.0425 


25 


2 


5 2 


63 


1.0450 


I-0475 


1.0500 


1.0524 


1.0549 


1 -0574 


25 


3 

4 

5 


78 


75 
10 

125 


64 


1.0598 


1.0623 


1.0648 


1.0672 


1.0697 


1. 0721 


25 


10 4 
13 


65 


1.0746 


1. 0771 


1 -0795 


1. 0819 


1.0844 


1.0868 


24 


66 


1.0893 


1. 0917 


1.0942 


1.0966 


1.0990 


1.1014 


24 


6 


156 


150 


67 


1.1039 


1 . 1063 


1. 1087 


1. mi 


1-1136 


1.1160 


24 


7 


18 2 


17 5 


68 


1.1184 


1. 1208 


1. 1232 


1. 1256 


1. 1280 


1. 1304 


24 


8 


20 8 


20 


69 


1. 1328 


I-I352 


1-1376 


1. 1400 


1. 1424 


1. 1448 


24 


9 


23 4 


225 
2 3 


70 


1. 1472 


1 -1495 


1.1519 


1 -1543 


1-1567 


i-i59o 


24 


24 . 


7i 


1.1614 


1. 1638 


1.1661 


1. 1685 


1. 1709 


1. 1732 


24 


1 


2 4 


23 
46 


72 


1. 1756 


1. 1779 


1. 1803 


1. 1826 


1. 1850 


1.1873 


23 


2 


48 


73 


1. 1896 


1. 1920 


i- 1943 


1. 1966 


1. 1 990 


1. 2013 


23 


3 


7 2 


69 


74 


1.2036 


1.2060 


1.2083 


1. 2106 


1. 2129 


1. 2152 


23 


4 
5 


96 
12 


92 
"5 


75 


1. 2175 


1. 2198 


1. 2221 


1.2244 


1.2267 


1.2290 


23 


76 


1-2313 


1.2336 


1-2359 


1.2382 


1.2405 


1.2428 


23 


6 


144 


138 


77 


1.2450 


1-2473 


1.2496 


1.2518 


1.2541 


1.2564 


23 


7 


168 


16 1 


78 


1.2586 


1.2609 


1.2632 


1.2654 


1.2677 


1.2699 


23 


8 


19 2 


18 4 


79 


1.2722 


1.2744 


1.2766 


1.2789 


1.2811 


1-2833 


22 


9 


21 6 


207 
21 


80 


1.2856 


1.2878 


1.2900 


1.2922 


1-2945 


1.2967 


22 


22 


81 


1.2989 


1.3011 


1-3033 


1-3055 


1.3077 


1.3099 


22 


1 


2 2 


2 1 


82 


1.3121 


i-3 J 43 


1-3165 


1-3187 


1.3209 


1-3231 


22 


2 


4 4 


4 2 


83 


1-3252 


1-3274 


1.3296 


i-33i8 


1-3339 


i-336i 


22 


3 


66 


63 

84 

105 


84 


1-3383 


I-3404 


1.3426 


1-3447 


I-3469 


1.3490 


22 


4 

5 


8 8 
11 


85 


I-35I2 


1-3533 


1-3555 


I-3576 


1-3597 


1-3619 


21 


86 


1.3640 


1. 3661 


1.3682 


1.3704 


I-3725 


1-3746 


21 


6 


13 2 


12 6 


87 


1.3767 


1.3788 


1.3809 


1-3830 


1-3851 


1.3872 


21 


7 


15 4 


147 


88 


I-3893 


I-39I4 


1-3935 


I-3956 


1-3977 


1-3997 


21 


8 


17 6 


168 


89 


1. 4018 


1.4039 


1.4060 


1.4080 


1.4101 


1. 4122 


21 


9 


19 8 


18 9 
19 


90 


1. 4142 


1.4163 


1.4183 


1.4204 


1.4224 


1.4245 


20 


20 


9i 


1.4265 


1.4285 


1.4306 


1.4326 


1.4346 


I-4367 


20 


1 


2 


1 9 


92 


I-4387 


1.4407 


1.4427 


1.4447 


1:4467 


1.4487 


20 


2 


4 


38 


93 


I-4507 


I-4527 


1-4547 


I-4567 


I-4587 


1.4607 


20 


3 


60 


5 7 


94 


1.4627 


1.4647 


1.4667 


1.4686 


1.4706 


1.4726 


20 


4 
5 


80 
10 


76 
95 


95 


1.4746 


I-4765 


1.4785 


1.4804 


1.4824 


1.4843 


20 


96 


1.4863 


r.4882 


1.4902 


1. 4921 


1. 4941 


1.4960 


19 


6 


12 


114 


97 


1.4979 


1.4998 


1.5018 


I-5037 


1-5056 


I-5075 


19 


7 


14 


13 3 


98 


1-5094 


I-5II3 


I-5I32 


I-5I5I 


1-5170 


1.5189 


19 


8 


16 


152 


99 


1.5208 


1.5227 


1.5246 


1-5265 


1-5283 


i-53°2 


19 


9 


18 


171 
17 


100 


1.5321 


1 -534Q 


1-5358 


1-5377 


1-5395 


I-54I4 


18 


18 


IOI 


1-5432 


I-545I 


1.5469 


1.5488 


i-55o6 


1-5525 


18 


x 


1 8 


1 7 


102 


1-5543 


i-556i 


1-5579 


1-5598 


1.5616 


I-5634 


18 


2 


36 


3 4 


103 


^5652 


1.5670 


1.5688 


1.5706 


1-5724 


1-5742 


18 


3 


5 4 


5 1 


104 


1.5760 


1-5778 


1-5796 


1-5814 


1-5832 


I-5849 


18 


4 

5 


72 
90 


68 
85 


105 


1.5867 


1-5885 


1.5902 


1.5920 


1-5938 


1-5955 


18 


106 


1-5973 


1.5990 


1.6008 


1.6025 


1.6042 


1.6060 


17 


6 


10 8 


10 2 


107 


1.6077 


1.6094 


1.6112 


1. 6129 


1. 6146 


1. 6163 


17 


7 


12 6 


119 


108 


1. 6180 


1. 6197 


1. 6214 


1. 623 1 


1.6248 


1.6265 


17 


8 


144 


136 


109 


1.6282 


1.6299 


1. 6316 


1-6333 


1-6350 


1.6366 


17 


9 


16 2 


15 3 
15 14 


no 


1-6383 


1.6400 


1. 6416 


1-6433 


1.6450 


1.6466 


17 


16 


in 


1.6483 


1.6499 


1-6515 


1-6532 


1.6548 


1.6564 


16 


1 


16 


15 14 


112 


1.6581 


1-6597 


1. 6613 


1.6629 


1.6646 


1.6662 


16 


2 


3 2 


3° 28 


"3 


1.6678 


1.6694 


1. 6710 


1.6726 


1.6742 


1.6758 


16 


3 


48 


45 42 


114 


1.6773 


1.6789 


1.6805 


1. 6821 


1.6836 


1.6852 


16 


4 

5 


64 
80 


60 56 
75 70 


115 


1.6868 


1.6883 


1.6899 


1. 6915 


1.6930 


1 .6946 


16 


116 


1. 6961 


1.6976 


1.6992 


1.7007 


1.7022 


1.7038 


15 


6 


96 


90 84 


117 


I-7053 


1.7068 


1.7083 


1.7098 


1.7113 


1.7128 


15 


7 


112 10 5 98 


118 


I-7I43 


1-7158 


I-7I73 


1.7188 


1.7203 


1. 7218 


15 


8 


12 8 12 11 2 


119 


1-7233 


1.7247 


1.7262 


1.7277 


1. 7291 


1.7306 


15 


9 


14 4 13 5 12 6 




0' 


10' 


20' 


30' 


40' 


5o' 


d. 


P. 


P. 



STADIA TABLES 



155 



TABLE XVI 

Stadia Tables 1 



M. 


c 





] 





2 





3 







hor. dist. 


diff. elev. 


hor. dist. 


diff. elev. 


hor. dist. 


diff. elev. 


hor. dist. 


diff. elev. 


o' 


IOO.OO 


O.OO 


99-97 


I.74 


99.88 


3-49 


99-73 


5.23 


2 


IOO.OO 


O.06 


99-97 


I.80 


99.87 


3.55 


99.72 


5.28 


4 


IOO.OO 


O.I2 


99-97 


1.86 


99.87 


3.60 


99.71 


5-34 


6 


IOO.OO 


O.17 


99.96 


1.92 


99.87 


3.66 


99.71 


540 


8 


IOO.OO 


O.23 


99.96 


1.98 


99.86 


3-72 


99.70 


5-46 


IO 


IOO.OO 


O.29 


99.96 


2.04 


99.86 


3-78 


99.69 


5-5 2 


12 


IOO.OO 


o-35 


99.96 


2.09 


99.85 


3-84 


99.69 


5-57 


H 


IOO.OO 


0.41 


99-95 


2.15 


99.85 


3.9o 


99.68 


5-63 


16 


IOO.OO 


047 


99-95 


2.21 


99.84 


3.95 


99.68 


5-69 


18 


IOO.OO 


0.52 


99-95 


2.27 


99.84 


4.01 


99.67 


5-75 


20 


IOO.OO 


0.58 


99-95 


2-33 


99.83 


4.07 


99.66 


5.80 


22 


IOO.OO 


0.64 


99-94 


2.38 


99.83 


4-13 


99.66 


5.86 


24 


IOO.OO 


0.70 


99-94 


2.44 


99.82 


4.18 


99.65 


5-92 


26 


99.99 


0.76 


99-94 


2.50 


99.82 


4.24 


99.64 


5-98 


28 


99.99 


0.81 


99-93 


2.56 


99.81 


4-3° 


99.63 


6.04 


30 


99.99 


0.87 


99-93 


2.62 


99.81 


4.36 


99.63 


6.09 


32 


99-99 


0-93 


99-93 


2.67 


99.80 


4.42 


99.62 


6.15 


34 


99.99 


0.99 


99-93 


2-73 


99.80 


4.48 


99.62 


6.21 


36 


99.99 


1.05 


99.92 


2.79 


9979 


4-53 


99.61 


6.27 


38 


99.99 


1. 11 


99.92 


2.85 


99-79 


4-59 


99.60 


6-33 


40 


99.99 


1. 16 


99.92 


2.91 


99.78 


4-65 


99-59 


6.38 


42 


99.99 


1.22 


99.91 


2-97 


99.78 


4.71 


99-59 


6.44 


44 


99.98 


1.28 


99.91 


3.02 


99-77 


4.76 


99-58 


6.50 


46 


99.98 


i-34 


99.90 


3.08 


99-77 


4.82 


99-57 


6.56 


48 


99.98 


1.40 


99.90 


3-14 


99.76 


4.88 


99-56 


6.61 


50 


99.98 


i-45 


99.90 


3.20 


99.76 


4.94 


99-56 


6.67 


52 


99.98 


r -5i 


99.89 


3.26 


99-75 


4-99 


99-55 


6-73 


54 


99.98 


i-57 


99.89 


3-31 


99-74 


5-°5 


99-54 


6.78 


56 


99-97 


1.63 


99.89 


3-37 


99-74 


5-ii 


99-53 


6.84 


58 


99-97 


1.69 


99.88 


3-43 


99-73 


5-i7 


99.52 


6.90 


60 
c = O.75 
c = 1. 00 
<: = I.25 


99-97 


1.74 


99.88 


3-49 


99-73 


5- 2 3 


99-5 1 


6.96 


°-75 


0.0 1 


°-75 


0.02 


o-75 


0.03 


°-75 


0.05 


1. 00 


O.OI 


1. 00 


0.03 


1. 00 


0.04 


1. 00 


0.06 


1.25 


0.02 


J - 2 5 


0.03 


1.25 


0.05 


1.25 


0.08 



1 These tables, copied here by permission, were computed by Mr. Arthur Winslow 
of the State Geological Survey of Pennsylvania. 



156 



STADIA TABLES 



TABLE XVI — Continued 
Stadia Tables 



M. 


4 





5 





6 





7 







hor. dist. 


diff. elev. 


hor. dist. 


diff. elev. 


hor. dist. 


diff. elev. 


hor. dist. 


diff. elev. 


o' 


99.51 


6.96 


99.24 


8.68 


98.91 


IO.40 


98.51 


I2.IO 


2 


99-51 


7.02 


99.23 


8.74 


98.90 


IO.45 


98.50 


12.15 


4 


99.5O 


7.O7 


99.22 


8.80 


98.88 


IO.51 


98.48 


12.21 


6 


99.49 


7.13 


99.21 


8.85 


98.87 


IO-57 


98.47 


12.26 


8 


99.48 


7.19 


99.20 


8.91 


98.86 


IO.62 


98.46 


12.32 


IO 


9947 


7.25 


99.19 


8,97, 


98.85 


IO.68 


98.44 


I2.38 


12 


99.46 


7-30 


99.18 


9-03 


98.83 


IO.74 


98.43 


12.43 


14 


99.46 


7.36 


99.17 


9.08 


98.82 


IO.79 


98.4I 


12.49 


16 


99-45 


742 


99.16 


9.I4 


98.81 


IO.85 


98.40 


12-55 


18 


99.44 


7.48 


99-15 


9.20 


98.80 


IO.91 


98.39 


I2.6o 


20 


9943 


7-53 


99.I4 


9.25 


98.78 


IO.96 


98.37 


12.66 


22 


99.42 


7-59 


99-13 


9.31 


98.77 


II.02 


98.36 


12.72 


24 


99.4I 


7-65 


99.II 


9-37 


98.76 


II.08 


98.34 


12.77 


26 


99.4O 


7.71 


99.IO 


9-43 


98.74 


II. 13 


98.33 


12.83 


28 


99-39 


7.76 


99.09 


948 


98.73 


II. 19 


98.31 


12.88 


30 


99-38 


7.82 


99.08 


9-54 


98.72 


II.25 


98.29 


12.94 


32 


99-38 


7.88 


99.07 


9.60 


98.71 


II.30 


98.28 


13.00 


34 


99-37 


7-94 


99.06 


9-65 


98.69 


11.36 


98.27 


13.05 


36 


99.36 


7-99 


99.05 


9.71 


98.68 


II.42 


98.25 


13-11 


38 


99-35 


8.05 


99.04 


9-77 


98.67 


II.47 


98.24 


13.17 


40 


99-34 


8.11 


99-03 


-9.83 


98.65 


n-53 


98.22 


13.22 


42 


99-33 


8.17 


99.OI 


9.88 


98.64 


n-59 


98.20 


13.28 


44 


99-32 


8.22 


99.00 


9-94 


98.63 


11.64 


98.I9 


13.33 


46 


99.31 


8.28 


98.99 


10.00 


98.61 


11.70 


98.17 


13-39 


48 


99-30 


8.34 


98.98 


10.05 


98.60 


11.76 


98.16 


13-45 


5° 


99.29 


8.40 


98.97 


10. 1 1 


98.58 


11.81 


98.I4 


13.5° 


52 


99.28 


8.45 


98.96 


10.17 


98.57 


11.87 


98.13 


13.56 


54 


99.27 


8.51 


98.94 


10.22 


98.56 


n-93 


98.II 


13.61 


56 


99.26 


8.57 


98.93 


10.28 


98.54 


11.98 


98.IO 


13.67 


58 


99-25 


8.63 


98.92 


10.34 


98.53 


12.04 


98.08 


1373 


60 
c = 0.75 
<: = I. OO 
c = 1.25 


99.24 


8.68 


98.9I 


10.40 


98.51 


12.10 


98.06 


13.78 


o-75 


0.06 


0.75 


0.07 


0.75 


0.08 


0.74 


O.IO 


1. 00 


0.08 


O.99 


0.09 


0.99 


0.1 1 


0.99 


0.13 


1.25 


O.IO 


I.24 


0.1 1 


I.24 


0.14 


I.24 


0.16 



STADIA TABLES 



15T 



TABLE XVI — Continued 
Stadia Tables 



M. 


8 





g 





IO° 


ii° 




hor. dist. 


diff. elev. 


hor. dist. 


diff. elev. 


hor. dist. 


diff. elev. 


hor. dist. 


diff. elev. 


o' 


98.06 


13.78 


97-55 


15-45 


96.98 


I7.IO 


96.36 


18.73 


2 


98.05 


I3-84 


97-53 


I5-5 1 


96.96 


17.16 


96.34 


18.78 


4 


98.03 


I3.89 


97-52 


J 5-5 6 


96.94 


17.21 


96.32 


18.84 


6 


98.OI 


13-95 


97-5° 


15.62 


96.92 


17.26 


96.29 


18.89 


8 


98.OO 


I4.OI 


97.48 


!5- 6 7 


96.90 


17-32 


96.27 


18.95 


IO 


97.98 


I4.06 


97.46 


J 5-73 


96.88 


17-37 


96.25 


I9.OO 


12 


97-97 


14.12 


97-44 


15.78 


96.86 


17-43 


96.23 


I9.O5 


14 


97-95 


I4.I7 


97-43 


15.84 


96.84 


17.48 


96.21 


i9.II 


16 


97-93 


14.23 


97.41 


15-89 


96.82 


17-54 


96.18 


I9.16 


18 


97.92 


14.28 


97-39 


15-95 


96.80 


17-59 


96.16 


19.21 


20 


97.90 


14-34 


97-37 


16.00 


96.78 


17.65 


96.14 


I9.27 


22 


97.88 


I4.4O 


97-35 


16.06 


96.76 


17.70 


96.12 


I9.32 


24 


97.87 


1445 


97-33 


16.11 


96.74 


17.76 


96.09 


19.38 


26 


97.85 


I4-5 1 


97-31 


16.17 


96.72 


17.81 


96.07 


19-43 


28 


97-83 


14.56 


97.29 


16.22 


96.70 


17.86 


96.05 


I9.48 


30 


97.82 


14.62 


97.28 


16.28 


96.68 


17.92 


96.03 


19-54 


32 


97.80 


14.67 


97.26 


16.33 


96.66 


17.97 


96.OO 


19.59 


34 


97.78 


14.73 


97.24 


16.39 


96.64 


18.03 


95-98 


I9.64 


36 


97.76 


14.79 


97.22 


16.44 


96.62 


18.08 


95-96 


I9.7O 


48 


97-75 


14.84 


97.20 


16.50 


96.60 


18.14 


95-93 


19-75 


40 


97-73 


14.90 


97.18 


l6 -55 


96.57 


18.19 


95-91 


I9.80 


42 


97.71 


14.95 


97.16 


16.61 


96.55 


18.24 


95-89 


19.86 


44 


97.69 


15.01 


97-14 


16.66 


96.53 


18.30 


95.86 


I9.9I 


46 


97.68 


15.06 


97.12 


16.72 


96.51 


18.35 


95-84 


I9.96 


48 


97.66 


15.12 


97.10 


16.77 


96.49 


18.41 


95.82 


20.02 


5° 


97.64 


15-17 


97.08 


16.83 


96.47 


18.46 


95-79 


2O.O7 


52 


97.62 


I5-23 


97.06 


16.88 


96.45 


18.51 


95-77 


20.I2 


54 


97.61 


15.28 


97.04 


16.94 


96.42 


18.57 


95-75 


20.I8 


56 


97-59 


15-34 


97.02 


16.99 


96.40 


18.62 


95-72 


20.23 


58 


97-57 


15.40 


97.00 


17-05 


96.38 


18.68 


95-7° 


20.28 


60 
' = 0.75 
c — 1. 00 
*= 1.25 


97-55 


15-45 


96.98 


17.10 


96.36 


18.73 


95-68 


20.34 


0.74 


0.1 1 


0.74 


0.12 


O.74 


0.14 


o.73 


O.I5 


0.99 


0.15 


0.99 


0.16 


O.98 


0.18 


0.98 


0.20 


1.23 


0.18 


1.23 


0.21 


I.23 


0.23 


1.22 


O.25 



158 



STADIA TABLES 



TABLE XVI — Continued 
Stadia Tables 



M. 


12° 


13° 


14° 


15° 




hor. dist. 


diff. elev. 


hor. dist. 


diff. elev. 


hor. dist. 


diff. elev. 


hor. dist. 


diff. elev. 


o' 


95-68 


20.34 


94.94 


21.92 


94.15 


2347 


93-30 


25.OO 


2 


95-65 


20.39 


94.91 


21.97 


94.12 


23.52 


93-27 


25. 5 


4 


95-63 


20.44 


94.89 


22.02 


94.09 


23.58 


93.24 


25.IO 


6 


95.61 


20.50 


94.86 


22.o8 


94.07 


23.63 


93.21 


25-I5 


8 


95-58 


20.55 


94.84 


22.13 


94.04 


23.68 


93.18 


25.20 


IO 


95-56 


20.60 


94.81 


22.18 


94.OI 


23.73 


93.16 


25-25 


12 


95-53 


20.66 


94-79 


22.23 


93.98 


2378 


93-13 


25-30 


H 


95-5 1 


20.71 


94.76 


22.28 


93-95 


23-83 


93.IO 


25-35 


16 


95-49 


20.76 


94-73 


22.34 


93-93 


23.88 


93-07 


25.40 


18 


9546 


20.81 


94.71 


22.39 


93-90 


23-93 


93-04 


2545 


20 


95-44 


20.87 


94.68 


22.44 


93.87 


23-99 


93.OI 


25-50 


22 


95-41 


20.92 


94.66 


22.49 


93.84 


24.04 


92.98 


25-55 


24 


95-39 


20.97 


94.63 


22.54 


93.8i 


24.09 


92.95 


25.60 


26 


95-36 


21.03 


94.60 


22.6o 


93-79 


24.14 


92.92 


25-65 


28 


95-34 


2I.08 


94.58 


22.65 


93-76 


24.19 


92.89 


25.70 


30 


95-32 


21.13 


94-55 


22.70 


93-73 


24.24 


92.86 


25-75 


32 


95-29 


2I.I8 


94.52 


22-75 


93-7o 


24.29 


92.83 


25.80 


34 


95-27 


21.24 


94-5° 


22.8o 


93-67 


24.34 


92.80 


25.85 


36 


95-24 


21.29 


94-47 


22.85 


93-65 


24-39 


92.77 


25.90 


38 


95-22 


21.34 


94-44 


22.91 


93.62 


24.44 


92.74 


25-95 


40 


95-J9 


21.39 


94.42 


22.96 


93-59 


24.49 


92.71 


26.OO 


42 


95-J7 


21.45 


94-39 


23.OI 


93-56. 


24-55 


92.68 


26.05 


44 


95- J 4 


21.50 


94.36 


23.06 


93-53 


24.60 


92.65 


26.IO 


46 


95.12 


21-55 


94-34 


23.II 


93-5° 


24.65 


92.62 


26.15 


48 


95-°9 


2I.6o 


94.31 


23.16 


9347 


24.70 


92.59 


26.20 


5° 


95-°7 


21.66 


94.28 


23.22 


9345 


24.75 


92.56 


26.25 


5 2 


95-°4 


21.71 


94.26 


23.27 


9342 


24.80 


92.53 


26.30 


54 


95.02 


21.76 


94-23 


23-32 


93-39 


24.85 


92.49 


26.35 


56 


94.99 


21.81 


94.20 


23-37 


93.36 


24.90 


92.46 


26.40 


58 


94-97 


21.87 


94.17 


23.42 


93-33 


24-95 


9243 


26.45 


60 
' = 0.75 

C — I. OO 

*= 1.25 


94.94 


21.92 


94-15 


23-47 


93-3° 


25.00 


92.40 


26.50 


o.73 


0.16 


o-73 


O.17 


o.73 


0.19 


O.72 


0.20 


0.98 


0.22 


0.97 


O.23 


o-97 


0.25 


O.96 


O.27 


1.22 


0.27 


1. 21 


O 29 


1. 21 


0.31 


I.20 


0.34 



STADIA TABLES 



159 



TABLE XVI — Continued 
Stadia Tables 



M. 


1 6° 


I 


7° 


I 


3° 


19° 




hor. dist. 


diff. elev. 


hor. dist. 


diff. elev. 


hor. dist. 


diff. elev. 


hor. dist. 


diff. elev. 


o' 


92.4O 


26.50 


91.45 


27.96 


90.45 


29-39 


89.40 


30.78 


2 


92.37 


26.55 


91.42 


28.01 


90.42 


29.44 


89.36 


30.83 


4 


92.34 


26.59 


91.39 


28.06 


90.38 


29.48 


89.32 


30.87 


6 


92.31 


26.64 


91-35 


28.IO 


90.35 


29-53 


89.29 


30.92 


8 


92.28 


26.69 


91.32 


28.15 


90.31. 


29.58 


89.26 


30.97 


IO 


92.25 


26.74 


91.29 


28.20 


90.28 


29.62 


89.22 


3I.OI 


12 


92.22 


26.79 


91.26 


28.25 


9O.24 


29.67 


89.18 


3I.06 


14 


92.19 


26.84 


91.22 


28.30 


90.2I 


29.72 


89.15 


3I.IO 


16 


92.15 


26.89 


91.19 


28.34 


90.18 


29.76 


89.II 


31.15 


18 


92.12 


26.94. 


91.16 


28.39 


9O.I4 


29.81 


89.08 


3 I - I 9 


20 


92.09 


26.99 


91.12 


28.44 


90.II 


29.86 


89.O4 


31.24 


22 


92.06 


27.04 


91.09 


28.49 


90.07 


29.90 


89.OO 


31.28 


24 


92.03 


27.09 


91.06 


28.54 


9O.O4 


29-95 


88.96 


31-33 


26 


92.OO 


27-!3 


91.02 


28.58 


90.OO 


30.OO 


88.93 


31.38 


28 


91.97 


27.18 


90.99 


28.63 


89.97 


30.04 


88.89 


3L42 


30 


91-93 


27.23 


90.96 


28.68 


89.93 


30.09 


88.86 


3147 


32 


9I.9O 


27.28 


90.92 


28.73 


89.90 


30.14 


88.82 


3I-5 1 


34 


91.87 


27-33 


90.89 


28.77 


89.86 


30.19 


88.78 


31-56 


36 


9I.84 


27-38 


90.86 


28.82 


89.83 


30.23 


88.75 


31.60 


38 


9I.81 


27-43 


90.82 


28.87 


89.79 


30.28 


88.71 


31.65 


40 


91.77 


27.48 


90.79 


28.92 


89.76 


30.32 


88.67 


31.69 


42 


91.74 


27.52 


90.76 


28.96 


89.72 


30.37 


88.64 


31.74 


44 


9I.7I 


27-57 


90.72 


29.OI 


89.69 


30.41 


88.60 


31-78 


46 


9I.68 


27.62 


90.69 


29.06 


89.65 


30.46 


88.56 


31.83 


48 


91.65 


27.67 


90.66 


29.II 


89.61 


30.5I 


88.53 


3I-87 


50 


9I.6l 


27.72 


90.62 


29.15 


89.58 


30.55 


88.49 


31.92 


52 


91.58 


27.77 


90.59 


29.20 


89.54 


30.60 


88.45 


31.96 


54 


91-55 


27.81 


90-55 


29.25 


89.51 


30.65 


88.41 


32.01 


56 


91.52 


27.86 


90.52 


29.30 


89.47 


30.69 


88.38 


32.05 


58 


9I.48 


27.91 


90.48 


29-34 


89.44 


30.74 


88.34 


32.09 


60 
c = 0.75 
f = 1. 00 
c — I.25 


91.45 


27.96 


90.45 


29.39 


89.40 


30.78 


88.30 


32.14 


O.72 


0.21 


0.72 


O.23 


O.7I 


O.24 


0.71 


0.25 


O.96 


0.28 


o.95 


O.30 


0.95 


O.32 


o.94 


033 


I.20 


0.36 


1. 19 


O.38 


1. 19 


O.4O 


1. 18 


0.42 



160 



STADIA TABLES 



TABLE XVI — Continued 
Stadia Tables 



1 

M. 


20° 


21° 


22° 


23° 




hor. dist. 


diff. elev. 


hor. dist. 


diff. elev. 


hor. dist. 


diff. elev. 


hor. dist. 


diff. elev. 


o' 


88.30 


32.14 


87.16 


33-46 


85-97 


34-73 


84.73 


35-97 


2 


88.26 


32.18 


87.12 


33-50 


85.93 


34-77 


84.69 


36.OI 


4 


88.23 


32.23 


87.08 


33-54 


85.89 


34.82 


84.65 


36.05 


6 


88.19 


32.27 


87.04 


33-59 


85.85 


34-86 


84.61 


36.09 


8 


88.15 


32.32 


87.OO 


33.63 


85.80 


34-90 


84.57 


36.13 


IO 


88.11 


32.36 


86.96 


33.67 


85.76 


34-94 


84.52 


36.17 


12 


88.08 


32.41 


86.92 


33-72 


85.72 


34.98 


84.48 


36.21 


14 


88.04 


32-45 


86.88 


3376 


85.68 


35-02 


84.44 


36.25 


16 


88.00 


3249 


86.84 


33-8o 


85.64 


35-07 


84.40 


36.29 


18 


87.96 


32.54 


86.80 


33.84 


85.60 


35-11 


84.35 


36.33 


20 


87-93 


• 32.58 


86.77 


33-89 


85.56 


35- J 5 


84.31 


36.37 


22 


87.89 


32.63 


86.73 


33-93 


85.52 


35-19 


84.27 


36.4I 


24 


87.85 


32.67 


86.69 


33-97 


85.48 


35-23 


84.23 


3645 


26 


87.81 


32.72 


86.65 


34.01 


8544 


35-27 


84.18 


3649 


28 


87-77 


32.76 


86.61 


34.06 


85.40 


35-31 


84.14 


36.53 


30 


87.74 


32.80 


86.57 


34.10 


85.36 


35.36 


84.IO 


36.57 


32 


87.70 


32.85 


86.53 


34-14 


85.31 


3540 


84.06 


36.61 


34 


87.66 


32.89 


86.49 


34.18 


85.27 


35-44 


84.OI 


36.65 


36 


87.62 


32.93 


86.45 


34-23 


85.23 


3548 


83-97 


36.69 


38 


87.58 


32.98 


86.41 


34-27 


85.19 


35-52 


83-93 


36.73 


40 


87.54 


33.02 


86.37 


34.31 


85.15 


35-56 


83.89 


36.77 


42 


87.51 


33-07 


86.33 


34-35 


85.II 


35-6o 


83.84 


36.80 


44 


87-47 


33-11 


86.29 


3440 


85.07 


35-64 


83.80 


36.84 


46 


8743 


33-15 


86.25 


3444 


85.02 


35-68 


83.76 


36.88 


48 


87.39 


33-20 


86.21 


3448 


84.98 


35-72 


83.72 


36.92 


50 


87-35 


33.24 


86.17 


34-52 


84.94 


3576 


83.67 


37.96 


52 


87-3I 


33-28 


86.13 


34-57 


84.90 


35.80 


83.63 


37.00 


54 


87.27 


33-33 


86.09 


34.6i 


84.86 


35-85 


83.59 


37.04 


5 S 


87.24 


33-37 


86.05 


34.65 


84.82 


35.89 


83.54 


37-08 


58 


87.20 


33-41 


86.01 


34.69 


8477 


35-93 


83.50 


37- 12 


60 
c = 0.75 

<T = I.OO 

c — 1.25 


87.16 


3346 


85-97 


34-73 


84-73 


35-97 


8346 


37-!6 


0.70 


0.26 


0.70 


0.27 


O.69 


0.29 


O.69 


0.30 


0.94 


o.35 


o.93 


o.37 


O.92 


0.38 


O.92 


0.40 


1. 17 


044 


1. 16 


0.46 


I.I5 


0.48 


i.!5 


0.50 



STADIA TABLES 



161 



TABLE XVI — Continued 
Stadia Tables 



M. 



24. 



25 



26 c 



27- 



22 
24 
26 
28 
30 

32 

34 
36 
38 
40 

42 

44 
46 
48 
50 

52 
54 

5 £ 
58 
60 



' = o-75 



c — 1.25 



hor. dist. 
8346 
83.41 
83.37 

^3-33 
83.28 

83.24 

83.20 

83.15 
83.11 
83.07 
83.02 

82.98 

82.93 
82.89 
82.85 
82.80 

82.76 
82.72 
82.67 
82.63 
82.58 

82.54 

82.49 
82.45 
82.41 
82.36 

82.32 
82.27 
82.23 
82.18 
82.14 



0.68 



0.91 



H 



diff. elev. 
37-16 
37.20 
37-23 
37.27 
37-31 

37-35 

37-39 
37-43 
37-47 
37-5 1 
37-54 

37-58 
37.62 
37.66 
37-7° 
37-74 

37-77 
37-8i 
37-85 
37-89 
37-93 

37-96 
38.00 
38.04 
38.08 
38.11 

38.15 
38.19 
38.23 
38.26 
38.30 



0.31 



hor. dist. 
82.I4 
82.09 
82.05 
82.OI 
81.96 
81.92 

81.87 
81.83 
81.78 
81.74 
81.69 

81.65 
8l.6o 
81.56 
81.51 
81.47 

81.42 

81.38 

81.33 
81.28 
81.24 

8I.I9 
81.15 
8I.IO 
81.06 
8I.OI 

80.97 
80.92 
80.87 
80.83 
80.78 



O.68 



O.4I 



O.52 



O.9O 



13 



diff. elev. 

38.30 

38.34 
38.38 
38.41 
38.45 
38.49 

38.53 
38.56 
38.60 
38.64 
38.67 

38.71 
38.75 
38.78 
38.82 
38.86 

38.89 
38.93 
38.97 
39.OO 

39-04 
39.08 

39-n 
39-15 
39.18 
39.22 

39.26 
39-29 
39-33 
39-36 
39-40 



0.32 



0.43 



o.54 



hor. dist. 
80.78 
80.74 
80.69 
80.65 
80.60 
80.55 

80.51 
80.46 
80.4I 

80.37 
80.32 

80.28 
80.23 
80.18 
80.14 
80.09 

80.04 
80.OO 
79-95 

79-Qo 
79.86 

79.81 
79.76 

7972 
79.67 
79.62 

79.58 
79-53 
79.48 

79-44 
79-39 



0.67 



0.89 



diff. elev. 
39.40 
39-44 

39-47 
39-5 1 
39-54 
39.58 

39.61 
39.65 
39.69 
39-72 
39.76 

39-79 
39.83 
39-86 
39.9o 
39-93 

39-97 
40.00 
40.04 
40.07 
40.11 

40.14 
40.18 
40.21 
40.24 
40.28 

40.31 

40.35 
40.38 
40.42 

4045 



o.33 



hor. dist. 

79-39 
79-34 
79-3o 
79.25 
79.20 

79.15 

79.11 
79.06 
79.01 
78.96 
78.92 

78.87 
78.82 
78.77 

78.73 
78.68 

78.63 

78.58 
78.54 
78.49 
78.44 

78.39 

78.34 
78.30 
78.25 
78.20 

78.15 
78.10 
78.06 
78.01 
77.96 



0.66 



•45 



0.56 



diff. elev. 

40.45 
40.49 
40.52 
40.55 
40.59 
40.62 

40.66 
40.69 
40.72 
40.76 
40.79 

40.82 
40.86 
40.89 
40.92 
4O.96 

40.99 
4I.02 
4I.06 
41.09 
41.12 

4I.16 
4I.I9 
41.22 
41.26 
4I.29 

4I.32 
41.35 
41.39 
4I.42 

41-45 



0.35 



O.46 



O.58 



162 



STADIA TABLES 







TABLE 


XVI— Concluded 










Stadia Tables 






M. 


28° 


29 


3°° 




hor. dist. 


diff. elev. 


hor. dist. 


diff. elev. 


hor. dist. 


diff. elev. 


o' 


77.96 


41.45 


76.50 


42.40 


75.OO 


43-30 


2 


77.91 


41.48 


76.45 


4243 


74-95 


43-33 


4 


77.86 


41.52 


76.4O 


42.46 


74.90 


43-36 


6 


77.81 


41.55 


76.35 


42.49 


74.85 


43-39 


8 


77-77 


41.58 


76.30 


42.53 


74.80 


43-42 


IO 


77.72 


4I.61 


76.25 


42.56 


74-75 


43-45 


12 


77.67 


41.65 


76.20 


42.59 


74.70 


43-47 


14 


77.62 


41.68 


76.15 


42.62 


74-65 


43-5° 


16 


77-57 


4I.7I 


76.IO 


42.65 


74.60 


43-53 


18 


77-52 


41.74 


76.05 


42.68 


74-55 


43.56 


20 


7748 


41.77 


76.OO 


42.71 


74-49 


43-59 


22 


77-42 


4I.81 


75-95 


42.74 


74-44 


43.62 


24 


77.38 


41.84 


75-90 


42.77 


74-39 


43.65 


26 


77-33 


41.87 


75.85 


42.80 


74.34 


43.67 


28 


77.28 


41.90 


75.80 


42.83 


74.29 


43.70 


30 


77.23 


41-93 


75-75 


42.86 


74.24 


43-73 


32 


77.18 


41.97 


75.70 


42.89 


74.19 


43-76 


34 


77.13 


42.OO 


75.65 


42.92 


74.I4 


43-79 


36 


77.09 


42.03 


75.60 


42.95 


74.09 


43-82 


38 


77.04 


42.06 


75-55 


42.98 


74.O4 


43.84 


40 


76.99 


42.09 


75.50 


43.OI 


73-99 


43-87 


42 


76.94 


42.12 


75-45 


43-°4 


73-93 


43.90 


44 


76.89 


42.15 


75-40 


43-07 


73.88 


43-93 


46 


76.84 


42.19 


75-35 


43.IO 


73.83 


43-95 


48 


76.79 


42.22 


75-3o 


43-13 


7378 


43-98 


50 


76.74 


42.25 


75-25 


43.16 


73-73 


44.01 


52 


76.69 


42.28 


75.20 


43.18 


73.68 


44.04 


54 


76.64 


42.31 


75-15 


43.21 


73.63 


44.07 


56 


76.59 


42.34 


75- IQ 


43-24 


73.58 


44.09 


58 


76.55 


42.37 


75-05 


43-27 


73.52 


44.12 


60 

c = 0.75 
<: = I.OO 

c — 1.25 


76.50 


42.40 


75.00 


43-30 


73-47 


44-15 


0.66 


O.36 


0.65 


0-37 


0.65 


0.38 


0.88 


O.48 


0.87 


0.49 


0.86 


0.51 


1. 10 


O.60 


1.09 


0.62 


1.08 


0.64 



TABLE XVII 



LOGARITHMS OF NUMBERS 



FROM 1 TO 10,000. 



164 



LOGARITHMS OF NUMBERS. 



N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 


IOO 


00 0000 


00 0434 


00 0868 


00 1301 


001734 


00 2166 


00 2598 


00 3029 


00 3461 


00 3891 


432 


IOI 


4321 


475i 


5181 


5609 


6038 


6466 


6894 


7321 


7748 


8174 


428 


102 


8600 


9026 


945i 


9876 


01 0300 


01 0724 


01 1 147 


01 1570 


01 1993 


01 2415 


424 


I03 


01 2837 


01 3259 


01 3680 


01 4100 


4521 


4940 


536o 


5779 


6197 


6616 


420 


IO4 


7°33 


745i 


7868 


8284 


8700 


9116 


9532 


9947 


02 0361 


020775 


416 


I05 


02 1 189 


02 1603 


02 2016 


02 2428 


02 2841 


02 3252 


02 3664 


02 4075 


02 4486 


02 4896 


412 


I06 


53o6 


57i5 


6125 


6533 


6942 


735° 


7757 


8164 


8571 


8978 


408 


107 


9384 


9789 


030195 


030600 


03 1004 


03 1408 


03 1812 


03 2216 


03 2619 


03 3021 


404 


1 08 


03 3424 


03 3826 


4227 


4628 


5029 


543o 


5830 


6230 


6629 


7028 


400 


IOg 


7426 


7825 


8223 


8620 


9017 


94i4 


981 1 


04 0207 


04 0602 


04 0998 


397 


no 


04 1393 


041787 


04 2182 


042576 


04 2969 


04 3362 


°4 3755 


04 4148 


04 4540 


044932 


393 


III 


5323 


57i4 


6105 


6495 


6885 


7275 


7664 


8053 


8442 


8830 


39o 


112 


9218 


9606 


9993 


05 0380 


05 0766 


051153 


05 1538 


05 1924 


05 2309 


05 2694 


386 


113 


05 3078 


05 3463 


05 3846 


4230 


4613 


4996 


5378 


576o 


6142 


6524 


383 


114 


6905 


7286 


7666 


8046 


8426 


8805 


9185 


95 6 3 


9942 


06 0320 


379 


115 


06 0698 


06 1075 


06 1452 


06 1829 


06 2206 


06 2582 


06 2958 


06 3333 


06 3709 


06 4083 


376 


116 


445 8 


4832 


5206 


5580 


5953 


6326 


6699 


7071 


7443 


7815 


373 


117 


8186 


8557 


8928 


9298 


9668 


07 0038 


07 0407 


070776 


07 1 145 


07 15H 


37° 


118 


07 1882 


07 2250 


07 2617 


07 2985 


07 3352 


37i8 


4085 


445 ! 


4816 


5182 


366 


119 


5547 


5912 


6276 


6640 


7004 


7368 


773i 


8094 


8457 


8819 


363 


120 


07 9181 


07 9543 


07 9904 


08 0266 


08 0626 


08 0987 


08 1347 


08 1707 


08 2067 


08 2426 


360 


121 


08 2785 


08 3144 


08 3503 


3861 


4219 


4576 


4934 


5291 


5647 


6004 


357 


122 


6360 


6716 


7071 


7426 


7781 


8136 


8490 


8845 


9198 


9552 


355 


123 


9905 


090258 


09 061 1 


09 0963 


091315 


09 1667 


09 2018 


09 2370 


09 2721 


093071 


352 


124 


09 3422 


3772 


4122 


4471 


4820 


5169 


55i8 


5866 


6215 


6562 


349 


125 


09 6910 


097257 


09 7604 


097951 


09 8298 


09 8644 


09 8990 


09 9335 


09 9681 


100026 


346 


126 


10 0371 


100715 


10 1059 


10 1403 


10 1747 


10 2091 


10 2434 


102777 


103119 


3462 


343 


I27 


3804 


4146 


4487 


4828 


5 l6 9 


55io 


5851 


6191 


6531 


6871 


34i 


128 


7210 


7549 


7888 


8227 


8565 


8903 


9241 


9579 


99i6 


11 0253 


338 


129 


11 0590 


11 0926 


11 1263 


11 1599 


11 1934 


11 2270 


1 1 2605 


11 2940 


11 3275 


3609 


335 


130 


11 3943 


11 4277 


11 461 1 


1 1 4944 


11 5278 


11 5611 


11 5943 


11 6276 


11 6608 


11 6940 


333 


131 


7271 


7603 


7934 


8265 


8595 


8926 


9256 


9586 


9915 


120245 


33o 


132 


120574 


120903 


12 1231 


12 1560 


12 1888 


12 2216 


122544 


12 2871 


12 3198 


3525 


328 


133 


3852 


4178 


4504 


4830 


5i56- 


548i 


5806 


6131 


6456 


6781 


325 


134 


7105 


7429 


7753 


8076 


8399 


8722 


9045 


9368 


9690 


13 0012 


323 


135 


13 0334 


130655 


130977 


13 1298 


13 1619 


13 1939 


13 2260 


132580' 


13 2900 


13 3219 


321 


136 


3539 


3858 


4177 


4496 


4814 


5i33 


545 ! 


5769 


6086 


6403 


3i8 


137 


6721 


7037 


7354 


7671 


7987 


8303 


8618 


8934 


9249 


9564 


316 


138 


9879 


14 0194 


14 0508 


140822 


141136 


14 1450 


14 1763 


14 2076 


142389 


142702 


3H 


139 


14 3015 


3327 


3639 


3951 


4263 


4574 


4885 


5196 


5507 


5818 


311 


I40 


14 6128 


14 6438 


14 6748 


14 7058 


147367 


14 7676 


14 7985 


14 8294 


14 8603 


14 891 1 


309 


141 


9219 


9527 


9835 


15 0142 


15 0449 


150756 


15 1063 


15 i37o 


15 1676 


15 1982 


307 


142 


15 2288 


152594 


15 2900 


3205 


35io 


3815 


4120 


4424 


4728 


5032 


305 


143 


5336 


5640 


5943 


6246 


6549 


6852 


7i54 


7457 


7759 


8061 


303 


I44 


8362 


8664 


8965 


9266 


9567 


9868 


16 0168 


1 6 0469 


160769 


16 1068 


301 


145 


16 1368 


16 1667 


16 1967 


16 2266 


16 2564 


162863 


163161 


1 6 3460 


163758 


164055 


299 


I46 


4353 


4650 


4947 


5244 


554i 


5838 


6134 


6430 


6726 


7022 


297 


147 


7317 


7613 


79o8 


8203 


8497 


8792 


9086 


9380 


9674 


9968 


295 


I48 


17 0262 


170555 


170848 


171141 


17 1434 


17 1726 


17 2019 


172311 


172603 


172895 


293 


I49 


3186 


3478 


3769 


4060 


435 ! 


4641 


4932 


5222 


55!2 


5802 


291 


IS© 


17 6091 


17 6381 


17 6670 


17 6959 


177248 


17 7536 


177825 


178113 


17 8401 


178689 


289 


151 


8977 


9264 


9552 


9839 


180126 


180413 


180699 


180986 


18 1272 


18 1558 


287 


252 


18 1844 


18 2129 


18 2415 


18 2700 


2985 


3270 


3555 


3839 


4123 


4407 


285 


153 


4691 


4975 


5259 


5542 


5825 


6108 


6391 


6674 


6956 


7239 


283 


154 


7521 


7803 


8084 


8366 


8647 


8928 


9209 


9490 


9771 


19 0051 


281 


155 


190332 


190612 


190892 


191171 


I9H5 1 


19 1730 


19 2010 


19 2289 


192567 


19 2846 


279 


156 


3125 


3403 


3681 


3959 


4237 


45 H 


4792 


5069 


5346 


5623 


27S 


157 


5900 


6176 


6453 


6729 


7005 


7281 


7556 


7832 


8107 


8382 


276 


158 


8657 


8932 


9206 


9481 


9755 


20 0029 


20 0303 


200577 


20 0850 


20 1 1 24 


274 


159 


20 1397 


20 1670 


20 1943 


202216 


20 2488 


2761 


3033 


3305 


3577 


3848 


272 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 



LOGARITHMS OF NUMBERS. 



165 



N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 


160 


204120 


204391 


20 4663 


20 4934 


20 5204 


20 5475 


20 5746 


206016 


20 6286 


20 6556 


271 


161 


6826 


7096 


7365 


7634 


7904 


8173 


8441 


8710 


8979 


9247 


269 


162 


95*5 


9783 


21 0051 


21 0319 


210586 


21 0853 


21 1121 


21 1388 


21 1654 


21 1921 


267 


163 


21 2188 


21 2454 


2720 


2986 


3252 


3518 


3783 


4049 


43H 


4579 


266 


164 


4844 


5 io 9 


5373 


5638 


5902 


6166 


6430 


6694 


6957 


7221 


264 


165 


21 7484 


21 7747 


21 8010 


21 8273 


21 8536 


21 8798 


21 9060 


21 9323 


21 9585 


21 9846 


262 


166 


220108 


220370 


220631 


22 0892 


221153 


22 1414 


22 1675 


22 1936 


22 2196 


22 2456 


261 


167 


2716 


2976 


3236 


3496 


3755 


4015 


4274 


4533 


4792 


5051 


259 


168 


5309 


5568 


5826 


6084 


6342 


6600 


6858 


7"5 


7372 


7630 


258 


169 


7887 


8144 


8400 


8657 


8913 


9170 


9426 


9682 


9938 


230193 


256 


170 


23 0449 


23 0704 


23 0960 


23 1215 


23 1470 


23 1724 


23 1979 


23 2234 


23 2488 


23 2742 


255 


171 


2996 


3250 


3504 


3757 


401 1 


4264 


45i7 


477° 


5023 


5276 


253 


172 


5528 


578i 


6033 


6285 


6537 


6789 


7041 


7292 


7544 


7795 


252 


173 


8046 


8297 


8548 


8799 


9049 


9299 


955° 


9800 


24 0050 


24 0300 


250 


174 


24 0549 


240799 


24 1048 


24 1297 


24 1546 


24 1795 


24 2044 


24 2293 


2541 


2790 


249 


175 


24 3038 


24 3286 


24 3534 


24 3782 


24 4030 


24 4277 


24 4525 


244772 


245019 


24 5266 


248 


176 


55i3 


5759 


6006 


6252 


6499 


6745 


6991 


7237 


7482 


7728 


246 


177 


7973 


8219 


8464 


8709 


8954 


9198 


9443 


9687 


9932 


250176 


245 


178 


25 0420 


25 0664 


25 0908 


251151 


25 1395 


25 1638 


25 1881 


25 2125 


25 2368 


2610 


243 


179 


2853 


3096 


3338 


358o 


3822 


4064 


4306 


4548 


4790 


5031 


242 


180 


25 5273 


25 55H 


25 5755 


25 5996 


25 6237 


25 6477 


25 6718 


25 6958 


257198 


25 7439 


241 


181 


7679 


7918 


8158 


8398 


8637 


8877 


9116 


9355 


9594 


9833 


239 


182 


260071 


26 0310 


26 0548 


260787 


26 1025 


26 1263 


26 1501 


26 1739 


26 1976 


26 2214 


238 


183 


245 1 


2688 


2925 


3162 


' 3399 


3636 


3873 


4109 


4346 


4582 


237 


184 


4818 


5°54 


5290 


5525 


576i 


5996 


6232 


6467 


6702 


6937 


235 


185 


26 7172 


26 7406 


26 7641 


26 7875 


26 8110 


26 8344 


268578 


268812 


26 9046 


269279 


234 


186 


95 J 3 


9746 


9980 


270213 


27 0446 


270679 


27 0912 


27 1 144 


27 1377 


27 1609 


233 


187 


27 1842 


27 2074 


27 2306 


2538 


2770 


3001 


3233 


3464 


3696 


3927 


232 


188 


4158 


4389 


4620 


4850 


5081 


53" 


5542 


5772 


6002 


6232 


230 


189 


6462 


6692 


6921 


7151 


738o 


7609 


7838 


8067 


8296 


8525 


229 


190 


27 8754 


27 8982 


27 921 1 


27 9439 


27 9667 


27 9895 


28 0123 


280351 


28 0578 


28 0806 


228 


191 


28 1033 


28 1261 


28 1488 


281715 


28 1942 


28 2169 


2396 


2622 


2849 


3075 


227 


192 


330I 


3527 


3753 


3979 


4205 


4431 


4656 


4882 


5 IQ 7 


5332 


226 


193 


5557 


5782 


6007 


6232 


6456 


6681 


6905 


7130 


7354 


7578 


225 


194 


7802 


8026 


8249 


8473 


8696 


8920 


9143 


9366 


9589 


9812 


223 


195 


29 0035 


290257 


29 0480 


29 0702 


29 0925 


29 1 147 


29 i3 6 9 


29 1591 


29 1813 


29 2034 


222 


196 


2256 


2478 


2699 


2920 


3Hi 


33^3 


3584 


3804 


4025 


4246 


221 


197 


4466 


4687 


4907 


5 I2 7 


5347 


55 6 7 


5787 


6007 


6226 


6446 


220 


198 


6665 


6884 


7104 


7323 


7542 


7761 


7979 


8198 


8416 


8635 


219 


199 


8853 


9071 


9289 


9507 


9725 


9943 


300161 


30 0378 


300595 


300813 


218 


200 


30 1030 


3° I2 47 


30 1464 


30 1681 


30 1898 


302114 


302331 


30 2547 


30 2764 


30 2980 


217 


201 


3196 


3412 


3628 


3844 


4059 


4275 


4491 


4706 


4921 


5 J 36 


216 


202 


535i 


5566 


5781 


5996 


621 1 


6425 


6639 


6854 


7068 


7282 


215 


203 


7496 


7710 


7924 


8i37 


8351 


8564 


8778 


8991 


9204 


9417 


213 


204 


9630 


9843 


31 0056 


31 0268 


31 0481 


31 0693 


31 0906 


31 1118 


31 1330 


3i 1542 


212 


205 


3i 1754 


31 1966 


31 2177 


31 2389 


31 2600 


31 2812 


31 3023 


31 3234 


3 1 3445 


3i 3656 


211 


206 


3867 


4078 


4289 


4499 


4710 


4920 


5 J 3o 


5340 


555i 


5760 


210 


207 


597o 


6180 


6390 


6599 


6809 


7018 


7227 


7436 


7646 


7854 


209 


208 


8063 


8272 


8481 


8689 


8898 


9106 


93H 


9522 


973o 


9938 


208 


209 


320146 


320354 


320562 


320769 


320977 


32 1 184 


321391 


32 1598 


32 1805 


32 2012 


207 


210 


32 2219 


32 2426 


322633 


32 2839 


32 3046 


323252 


32 3458 


32 3 66 5 


323871 


32 4077 


206 


211 


4282 


4488 


4694 


4899 


5 io 5 


53 IQ 


55i6 


572i 


5926 


6131 


205 


212 


6336 


6541 


6745 


6950 


7*55 


7359 


7563 


7767 


7972 


8176 


204 


213 


8380 


8583 


8787 


8991 


9194 


9398 


9601 


9805 


33 0008 


33 021 1 


203 


214 


33 04H 


330617 


330819 


33 IQ 22 


33 1225 


33 1427' 


33 1630 


33 1832 


2034 


2236 


202 


215 


33 2438 


33 2640 


33 2842 


33 3°44 


33 3246 


33 3447 


33 3 6 49 


33 3850 


33 4051 


33 4253 


202 


216 


4454 


4655 


4856 


5°57 


5257 


5458 


5658 


5f59 


6059 


6260 


201 


217 


6460 


6660 


6860 


7060 


7260 


7459 


7659 


7858 


8058 


8257 


200 


218 


8456 


8656 


8855 


9054 


9253 


945 1 


9650 


9849 


34 0047 


34 0246 


199 


219 


34 0444 


34 0642 


340841 


34 io39 


341237 


34 1435 


34 1632 


34 1830 


2028 


2225 


198 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 



166 



LOGARITHMS OF NUMBERS. 



N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 


220 


34 2423 


34 2620 


342817 


34 30H 


343212 


34 3409 


34 3606 


34 3802 


34 3999 


344196 


197 


221 


4392 


4589 


4785 


4981 


5178 


5374 


557° 


5766 


5962 


6157 


196 


222 


6353 


6549 


6744 


6939 


7i35 


733o 


7525 


7720 


7915 


8110 


195 


223 


8305 


8500 


8694 


8889 


9083 


9278 


9472 


9666 


9860 


35 oo54 


194 


224 


35 02 48 


35 °442 


35 o6 36 


35 o82 9 


35 IQ2 3 


35 1216 


35 Hio 


35 1603, 


35 1796 


1989 


193 


225 


35 2183 


35 2375 


35 2568 


35 2 76i 


35 2 954 


35 3H7 


35 3339 


35 3532 


35 3724 


35 39i6 


r 93 


226 


4108 


4301 


4493 


4685 


4876 


5068 


5260 


5452 


5 6 43 


5834 


192 


227 


6026 


6217 


6408 


6599 


6790 


6981 


7172 


7363 


7554 


7744 


191 


228 


7935 


8125 


8316 


8506 


8696 


8886 


9076 


9266 


945 6 


9646 


190 


22g 


9835 


36 0025 


360215 


36 0404 


360593 


360783 


360972 


36 1 161 


361350 


36 1539 


189 


230 


361728 


361917 


36 2105 


36 2294 


36 2482 


362671 


36 2859 


36 3048 


36 3236 


36 3424 


188 


231 


3612 


3800 


3988 


4176 


4363 


455i 


4739 


4926 


5 IJ 3 


53 QI 


188 


232 


5488 


5675 


5862 


6049 


6236 


6423 


6610 


6796 


6983 


7169 


187 


233 


7356 


7542 


7729 


7915 


8101 


8287 


8473 


8659 


8845 


9030 


186 


234 


9216 


9401 


9587 


9772 


995 8 


370143 


37 0328 


370513 


370698 


370883 


185 


235 


37 Io68 


37 1253 


37 H37 


37 1622 


37 1806 


37 i99i 


37 2175 


37 2360 


37 2544 


37 2728 


184 


236 


2912 


3096 


3280 


3464 


3647 


3831 


4015 


4198 


4382 


4565 


184 


237 


4748 


4932 


5"5 


5298 


548i 


5664 


5846 


6029 


6212 


6394 


183 


238 


6577 


6759 


6942 


7124 


7306 


7488 


7670 


7852 


8034 


8216 


182 


239 


8398 


8580 


8761 


8943 


9124 


9306 


9487 


9668 


9849 


3^ 0030 


181 


240 


38 021 1 


38 0392 


380573 


380754 


38 0934 


381115 


38 1296 


38 1476 


38 1656 


38 1837 


181 


241 


2017 


2197 


2377 


2557 


2 737 


2917 


3097 


3277 


3456 


3636 


180 


242 


3815 


3995 


4174 


4353 


4533 


4712 


4891 


5070 


5249 


5428 


179 


243 


5606 


5785 


5964 


6142 


6321 


6499 


6677 


6856 


7034 


7212 


178 


244 


739o 


7568 


7746 


7923 


8101 


8279 


8456 


8634 


881 1 


8989 


178 


245 


389166 


38 9343 


389520 


38 9698 


38 9875 


390051 


39 0228 


39 0405 


39 0582 


39 0759 


177 


246 


39 0935 


391112 


391288 


39 H64 


39 1 641 


1817 


1993 


2169 


2345 


2521 


176 


247 


2697 


2873 


3048 


3224 


3400 


3575 


375 1 


3926 


4101 


4277 


176 


24S 


4452 


4627 


4802 


4977 


5^2 


5326 


55 QI 


5676 


5850 


6025 


*75 


249 


6199 


6374 


6548 


6722 


6896 


7071 


7245 


7419 


7592 


7766 


174 


250 


39 7940 


39 81 14 


39 8287 


39 8461 


39 8634 


39 8808 


398981 


39 9154 


39 9328 


39 95 QI 


l 73 


251 


9674 


9847 


40 0020 


400192 


40 0365 


400538 


40 07 1 1 


40 0883 


40 1056 


40 1228 


l 73 


252 


40 1401 


40 1573 


1745 


1917 


2089 


2261 


2433 


2605 


2777 


2949 


172 


253 


3121 


3292 


3464 


3635 


3807 


3978 


4149 


4320 


4492 


4663 


171 


254 


4834 


5oo5 


5^6 


5346 


5517 


5688 


5858 


6029 


6199 


6370 


171 


255 


40 6540 


406710 


406881 


407051 


40 7221 


407391 


40 7561 


40 7731 


40 7901 


40 8070 


170 


256 


8240 


8410 


8579 


8749 


8918 


9087 


9257 


9426 


9595 


9764 


169 


257 


9933 


41 0102 


41 0271 


41 0440 


41 0609 


41 0777 


41 0946 


41 1 1 14 


41 1283 


4i I45 1 


169 


258 


41 1620 


1788 


i95 6 


2124 


2293 


2461 


2629 


2796 


2964 


3 J 3 2 


168 


259 


33°° 


3467 


3635 


3803 


397o 


' 4i37 


4305 


4472 


4639 


4806 


167 


260 


4i 4973 


41 5140 


4i 5307 


4i 5474 


41 5641 


41 5808 


4i 5974 


41 6141 


41 6308 


41 6474 


167 


261 


6641 


6807 


6973 


7*39 


7306 


7472 


7638 


7804 


797° 


8i35 


166 


262 


8301 


8467 


8633 


8798 


8964 


9129 


9295 


9460 


9625 


9791 


165 


263 


9956 


420121 


420286 


420451 


420616 


420781 


42 0945 


42 1 1 10 


421275 


42 1439 


165 


264 


42 1604 


1768 


1933 


2097 


2261 


2426 


2590 


2754 


2918 


3082 


164 


265 


42 3246 


42 34io 


42 3574 


42 3737 


42 3901 


42 4065 


42 4228 


42 4392 


42 4555 


424718 


164 


266 


4882 


5045 


5208 


537 1 


5534 


5 6 97 


5860 


6023 


6186 


6349 


163 


267 


651 1 


6674 


6836 


6999 


7161 


7324 


7486 


7648 


781 1 


7973 


162 


268 


8i35 


8297 


8459 


8621 


^3 


8944 


9106 


9268 


9429 


959' 


162 


269 


9752 


9914 


43 oo75 


43 0236 


A3 0398 


430559 


43 0720 


43 0881 


43 1042 


43 ^03 


161 


270 


43 1364 


43 1525 


43 1685 


43 1846 


43 2007 


43 2167 


43 2328 


43 2488 


43 2649 


43 2809 


161 


271 


2969 


3130 


3290 


345° 


3610 


377o 


393o 


4090 


4249 


4409 


160 


272 


45 6 9 


4729 


4888 


5048 


5207 


5367 


5526 


5685 


5844 


6004 


*59 


273 


6163 


6322 


6481 


6640 


6799 


6 957 


7116 


7275 


7433 


7592 


l S9 


274 


7751 


7909 


8067 


8226 


8384 


8542 


8701 


8859 


9017 


9175 


158 


275 


43 9333 


43 9491 


43 9648 


43 9806 


43 9964 


440122 


44 0279 


44 0437 


440594 


440752 


158 


276 


44 0909 


44 1066 


44 1224 


441381 


44 1538 


1695 


1852 


2009 


2166 


2323 


157 


277 


2480 


2637 


2793 


2950 


3106 


3263 


3419 


3576 


3732 


3889 


157 


278 


4045 


4201 


4357 


4513 


4669 


4825 


4981 


5 J 37 


5293 


5449 


156 


279 


5604 


5760 


5915 


6071 


6226 


6382 


6537 


6692 


6848 


7003 


155 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 



LOGARITHMS OF NUMBERS. 



167 



N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 

*55 


280 


447158 


44 7313 


447468 


44 7 62 3 


44 7778 


44 7933 


44 8088 


44 8242 


448397 


448552 


281 


8706 


8861 


9015 


9170 


9324 


9478 


9633 


9787 


9941 


45°°95 


*54 


282 


45 ° 2 49 


45 °403 


45°557 


45 7*i 


45 0865 


45 1018 


45**72 


45 1326 


45 *479 


1633 


*54 


283 


1786 


1940 


2093 


2247 


2400 


2553 


2706 


2859 


3012 


3*65 


*53 


284 


33i8 


3471 


3624 


3777 


393° 


4082 


4 2 35 


4387 


4540 


4692 


*53 


285 


45 4845 


45 4997 


45 5*5° 


45 5302 


45 5454 


45 5606 


45 5758 


45 59*o 


45 6062 


45 6214 


*5 2 


286 


6366 


6518 


6670 


6821 


6973 


7125 


7276 


7428 


7579 


773* 


*5 2 


287 


7882 


8033 


8184 


8336 


8487 


8638 


8789 


8940 


9091 


9242 


*5* 


288 


9392 


9543 


9694 


9845 


9995 


460146 


460296 


460447 


460597 


46 0748 


*5* 


289 


46 0898 


46 1048 


46 1 198 


46 1348 


46 1499 


1649 


1799 


1948 


2098 


2248 


150 


290 


46 2398 


46 2548 


46 2697 


46 2847 


46 2997 


463146 


46 3296 


46 3445 


46 3594 


46 3744 


*5° 


291 


3893 


4042 


4191 


4340 


4490 


4639 


4788 


4936 


5085 


5234 


149 


292 


5383 


5532 


5680 


5829 


5977 


6126 


6274 


6423 


6571 


67*9 


*49 


293 


6868 


7016 


7164 


7312 


7460 


7608 


7756 


7904 


8052 


8200 


148 


294 


8347 


8495 


8643 


8790 


8938 


9085 


9 2 33 


938o 


9527 


9675 


148 


295 


46 9822 


46 9969 


470116 


47 0263 


470410 


470557 


47 0704 


470851 


47 0998 


47**45 


147 


296 


47 1292 


47 H38 


1585 


1732 


1878 


2025 


2171 


2318 


2464 


2610 


146 


297 


2756 


2903 


3049 


3*95 


334i 


3487 


3633 


3779 


3925 


4071 


146 


298 


4216 


4362 


4508 


4653 


4799 


4944 


5090 


5235 


538i 


5526 


146 


299 


5671 


5816 


5962 


6107 


6252 


6397 


6542 


6687 


6832 


6976 


*45 


300 


477121 


47 7266 


477411 


47 7555 


47 7700 


47 7844 


47 7989 


47 8i33 


478278 


47 8422 


*45 


301 


8566 


8711 


8855 


8999 


9143 


9287 


943* 


9575 


97*9 


9863 


144 


302 


48 0007 


48 01 5 1 


48 0294 


48 0438 


480582 


480725 


48 0869 


48 1012 


48 1 156 


48 1299 


144 


303 


1443 


1586 


1729 


1872 


2016 


2159 


2302 


2445 


2588 


2731 


*43 


304 


2874 


3016 


3159 


3302 


3445 


3587 


373o 


3872 


4015 


4*57 


*43 


305 


48 4300 


48 4442 


484585 


48 4727 


48 4869 


48 501 1 


485*53 


48 5295 


48 5437 


48 5579 


142 


306 


57 21 


5863 


6005 


6147 


6289 


6430 


6572 


6714 


6855 


6997 


142 


307 


7138 


7280 


7421 


7563 


7704 


7845 


7986 


8127 


8269 


8410 


141 


308 


855i 


8692 


8S33 


8974 


9114 


9255 


9396 


9537 


9677 


9818 


141 


309 


9958 


49 °°99 


49 0239 


49 38o 


49 0520 


49 0661 


49 0801 


49 0941 


49 108 1 


49 1222 


140 


310 


49 1362 


49 *5° 2 


49 1642 


491782 


49 1922 


49 2062 


49 2201 


49 2341 


49 2481 


49 2621 


140 


311 


2760 


2900 


3040 


3*79 


3319 


3458 


3597 


3737 


3876 


4015 


*39 


312 


4155 


4294 


4433 


457 '2 


471 1 


4850 


4989 


5128 


5267 


5406 


*39 


313 


5544 


5683 


5822 


5960 


6099 


6238 


6376 


65*5 


6653 


6791 


*39 


314 


6930 


7068 


7206 


7344 


7483 


7621 


7759 


7897 


8035 


8173 


138 


315 


49 831 1 


49 8448 


49 8586 


49 8724 


49 8862 


49 8999 


49 9*37 


49 9275 


499412 


49 955° 


138 


316 


9687 


9824 


9962 


50 0099 


500236 


500374 


50 05 1 1 


50 0648 


500785 


500922 


*37 


317 


5° io 59 


50 1196 


5 OI 333 


1470 


1607 


1744 


1880 


2017 


2154 


2291 


*37 


318 


2427 


2564 


2700 


2837 


2973 


3109 


3246 


3382 


35*8 


3655 


136 


319 


379i 


3927 


4063 


4199 


4335 


447 * 


4607 


4743 


4878 


50*4 


136 


320 


5° 5*5° 


50 5286 


5° 542i 


5° 5557 


5° 5 6 93 


505828 


5° 5964 


50 6099 


50 6234 


5° 6370 


136 


321 


6 5°5 


6640 


6776 


691 1 


7046 


7181 


73*6 


745* 


7586 


7721 


*35 


322 


7856 


7991 


8126 


8260 


8395 


8530 


8664 


8799 


8934 


9068 


*35 


323 


9203 


9337 


9471 


9606 


9740 


9874 


5 k 0009 


5*o*43 


5*0277 


51 041 1 


134 


324 


5 io 545 


510679 


51 0813 


5 1 °947 


51 1081 


51 1215 


1349 


1482 


1616 


* 75° 


*34 


325 


51 1883 


51 2017 


5 1 2I 5! 


51 2284 


51 2418 


5*255* 


51 2684 


51 2818 


5* 295* 


51 3084 


133 


326 


3218 


335 1 


3484 


3617 


375° 


3883 


4016 


4149 


4282 


4415 


*33 


327 


4548 


4681 


4813 


4946 


5°79 


5211 


5344 


5476 


5609 


574* 


*33 


328 


5874 


6006 


6139 


6271 


6403 


6535 


6668 


6800 


6932 


7064 


132 


329 


7196 


7328 


7460 


7592 


7724 


7855 


7987 


8119 


8251 


8382 


132 


330 


5*85*4 


51 8646 


5 l8 777 


51 8909 


51 9040 


5*917* 


5* 9303 


5* 9434 


5*9566 


519697 


*3* 


33i 


9828 


9959 


520090 


520221 


520353 


520484 


520615 


520745 


520876 


52 1007 


*3* 


332 


521138 


52 1269 


1400 


1530 


1661 


1792 


1922 


2053 


2183 


2314 


*3* 


333 


2444 


2575 


2705 


2835 


2966 


3096 


3226 


3356 


3486 


3616 


130 


334 


3746 


3876 


4006 


4*3 6 


4266 


4396 


4526 


4656 


4785 


49*5 


130 


335 


5 2 5°45 


5 2 5*74 


52 53°4 


52 5434 


525563 


5 2 5 6 93 


52 5822 


52 5951 


52 6081 


52 6210 


129 


336 


6339 


6469 


6598 


6727 


6856 


6985 


7**4 


7243 


7372 


75°* 


129 


337 


7630 


7759 


7888 


8016 


8145 


8274 


8402 


8531 


8660 


8788 


129 


338 


8917 


9045 


9174 


9302 


943o 


9559 


9687 


9815 


9943 


530072 


128 


339 


530200 


S3 0328 


53°45 6 


530584 


530712 


53 0840 


53 0968 


53 1096 


53 1223 


135* 


128 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 



168 



LOGARITHMS OF NUMBERS. 



N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 


340 


53 H79 


531607 


53 1734 


53 1862 


53 1990 


532117 


53 2245 


532372 


53 2500 


53 2627 


128 


341 


2754 


2882 


3009 


3136 


3264 


339i 


35i8 


3645 


3772 


3899 


127 


342 


4026 


4i53 


4280 


4407 


4534 


4661 


4787 


4914 


5041 


5^7 


127 


343 


5294 


5421 


5547 


5674 


5800 


5927 


6053 


6180 


6306 


6432 


126 


344 


6558 


6685 


681 1 


6937 


7063 


7189 


7315 


7441 


7567 


7693 


126 


345 


53 78i9 


53 7945 


538071 


53 8i97 


538322 


53 8448 


53 8574 


53 8699 


53 8825 


538951 


126 


346 


9076 


9202 


9327 


9452 


9578 


9703 


9829 


9954 


54 0079 


54 0204 


I2 5 


347 


540329 


54 0455 


540580 


540705 


540830 


54 0955 


54 -1080 


541205 


1330 


1454 


I2 5 


348 


1579 


1704 


1829 


1953 


2078 


2203 


2327 


2452 


2576 


2701 


125 


349 


2825 


2950 


3074 


3i99 


3323 


3447 


3571 


3696 


3820 


3944 


124 


350 


54 4068 


544192 


54 43i6 


54 444o 


54 4564 


54 4688 


544812 


54 4936 


54 5060 


54 5^3 


124 


35i 


5307 


543i 


5555 


5678 


5802 


5925 


6049 


6172 


6296 


6419 


124 


352 


6543 


6666 


6789 


6913 


7036 


7159 


7282 


7405 


7529 


7652 


123 


353 


7775 


7898 


8021 


8144 


8267 


8389 


8512 


8635 


8758 


8881 


123 


354 


9003 


9126 


9249 


937i 


9494 


9616 


9739 


9861 


9984 


550106 


123 


355 


55 0228 


55 0351 


55 0473 


55 °S95 


55 7i7 


55 0,840 


55 0962 


55 IQ 84 


55 1206 


55 1328 


122 


356 


145° 


1 57 2 


1694 


1816 


1938 


2060 


2181 


2303 


2425 


2547 


122 


357 


2668 


2790 


291 1 


3033 


3155 


3276 


3398 


35*9 


3640 


3762 


121 


358 


3883 


4004 


4126 


4247 


4368 


4489 


4610 


473i 


4852 


4973 


121 


359 


5°94 


5215 


5336 


5457 


5578 


5699 


5820 


594o 


6061 


6182 


121 


360 


55 6 303 


55 6423 


55 6544 


55 6664 


55 6785 


55 6905 


55 7026 


55 7146 


55 7267 


55 7387 


120 


361 


75°7 


7627 


7748 


7868 


7988 


8108 


8228 


8349 


8469 


8589 


120 


362 


8709 


8829 


8948 


9068 


9188 


9308 


9428 


9548 


9667 


9787 


120 


363 


9907 


56 0026 


560146 


560265 


560385 


56 0504 


560624 


56 0743 


56 0863 


56 0982 


119 


364 


56 IIOI 


1221 


^340 


1459 


1578 


1698 


1817 


1936 


2055 


2174 


119 


365 


562293 


56 2412 


562531 


56 2650 


56 2769 


562887 


56 3006 


563125 


56 3244 


56 3362 


119 


366 


348i 


3600 


37i8 


3837 


3955 


4074 


4192 


43i 1 


4429 


4548 


119 


367 


4666 


4784 


4903 


5021 


5i39 


5257 


5376 


5494 


5612 


573o 


118 


368 


5848 


5966 


6084 


6202 


6320 


6437 


6555 


6673 


6791 


6909 


118 


369 


7026 


7144 


7262 


7379 


7497 


7614 


7732 


7849 


7967 


8084 


118 


37o 


56 8202 


568319 


568436 


568554 


568671 


568788 


56 8905 


56 9023 


56 9140 


569257 


117 


37i 


9374 


9491 


9608 


9725 


9842 


9959 


570076 


57 OI 93 


570309 


57 0426 


117 


372 


57°543 


57 0660 


570776 


570893 


57 IOI ° 


57 1126 


1243 


1359 


1476 


1592 


117 


373 


1709 


1825 


1942 


2058 


2174 


2291 


2407 


2523 


2639 


2755 


116 


374 


2872 


2988 


3104 


3220 


3336 


3452 


3568 


3684 


3800 


3915 


116 


375 


57 4031 


57 4147 


574263 


574379 


57 4494 


574610 


574726 


57 4841 


57 4957 


57 5072 


116 


376 


5188 


5303 


54i9 


5534 


5650 


5765 


5880 


5996 


6111 


6226 


"5 


377 


6341 


6457 


6572 


6687 


6802 


6917 


7032 


7147 


7262 


7377 


"5 


378 


7492 


7607 


7722 


7836 


795 1 


8066 


8181 


8295 


8410 


8525 


"5 


379 


8639 


8754 


8868 


8983 


9097 


9212 


9326 


9441 


9555 


9669 


114 


380 


57 9784 


579898 


580012 


580126 


580241 


580355 


580469 


580583 


58 0697 


58 081 1 


114 


381 


580925 


58 1039 


1153 


1267 


1381 


1495 


1608 


1722 


1836 


1950 


114 


382 


2063 


2177 


2291 


2404 


2518 


2631 


2745 


2858 


2972 


3085 


114 


383 


3*99 


3312 


3426 


3539 


3652 


3765 


3879 


3992 


4105 


4218 


"3 


384 


4331 


4444 


4557 


4670 


4783 


4896 


5009 


5122 


5235 


5348 


113 


385 


585461 


58 5574 


58 5686 


5 8 5799 


585912 


58 6024 


586137 


58 6250 


58 6362 


58 6475 


113 


386 


6587 


6700 


6812 


6925 


7037 


7H9 


7262 


7374 


7486 


7599 


112 


387 


7711 


7823 


7935 


8047 


8160 


8272 


8384 


8496 


8608 


8720 


112 


388 


8832 


8944 


9056 


9167 


9279 


9391 


9503 


9615 


9726 


9838 


112 


389 


995° 


590061 


59oi73 


59 0284 


59 0396 


59 0507 


590619 


59 0730 


59 0842 


59 0953 


112 


390 


59 1065 


591176 


59 1287 


59 1399 


59 15 10 


59 J 62i 


59 1732 


59 1843 


59 1955 


59 2066 


in 


39i 


2177 


2288 


2399 


2510 


2621 


2732 


2843 


2954 


. 3064 


3175 


in 


392 


3286 


3397 


35o8 


3618 


3729 


3840 


395° 


4061 


4171 


4282 


in 


393 


4393 


4503 


4614 


4724 


4834 


4945 


5055 


5165 


5276 


5386 


no 


394 


5496 


5606 


5717 


5827 


5937 


6047 


6i57 


6267 


6377 


6487 


no 


395 


59 6597 


59 6707 


596817 


59 6927 


59 7037 


597146 


59 7256 


59 7366 


59 7476 


59 7586 


no 


396 


7695 


7805 


7914 


8024 


8i34 


8243 


*353 


8462 


8572 


8681 


no 


397 


8791 


8900 


9009 


9119 


9228 


9337 


9446 


9556 


9665 


9774 


109 


398 


9883 


9992 


600101 


600210 


600319 


60 0428 


600537 


60 0646 


600755 


60 0864 


109 


399 

N. 


600973 



60 1082 


1191 


1299 


1408 


!5 J 7 


1625 


1734 


1843 


I95 1 


109 
D. 


1 


2 


3 


4 


5 


6 7 


8 


9 



LOGARITHMS OF NUMBERS. 



169 



N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 


400 


60 2060 


60 2169 


60 2277 


60 2386 


60 2494 


60 2603 


60 271 1 


60 2819 


60 2928 


60 3036 


108 


401 


3H4 


3253 


3361 


3469 


3577 


3686 


3794 


3902 


4010 


4118 


108 


402 


4226 


4334 


4442 


455o 


4658 


4766 


4874 


4982 


5089 


5*97 


108 


403 


5305 


5413 


552i 


5628 


5736 


5 8 44 


595 1 


6059 


6166 


6274 


108 


404 


6381 


6489 


6596 


6704 


681 1 


6919 


7026 


7133 


7241 


7348 


107 


405 


60 7455 


60 7562 


60 7669 


60 7777 


60 7884 


60 7991 


60 8098 


60 8205 


608312 


60 8419 


107 


406 


8526 


8633 


8740 


8847 


8954 


9061 


9167 


9274 


9381 


9488 


107 


407 


9594 


9701 


9808 


9914 


61 0021 


61 0128 


61 0234 


61 0341 


61 0447 


61 0554 


107 


408 


61 0660 


61 0767 


61 0873 


61 0979 


1086 


1192 


1298 


1405 


J5 11 


1617 


106 


409 


1723 


1829 


1936 


2042 


2148 


2254 


2360 


2466 


2572 


2678 


106 


410 


61 2784 


61 2890 


61 2996 


61 3102 


61 3207 


61 3313 


61 3419 


61 3525 


61 3630 


61 3736 


106 


411 


3842 


3947 


4053 


4159 


4264 


437° 


4475 


4581 


4686 


4792 


106 


412 


4897 


5003 


5108 


5213 


5319 


5424 


5529 


5634 


5740 


5845 


io 5 


413 


5950 


6055 


6160 


6265 


6370 


6476 


6581 


6686 


6790 


6895 


io 5 


414 


7000 


7!°5 


7210 


73i5 


7420 


7525 


7629 


7734 


7839 


7943 


105 


415 


61 8048 


61 8153 


61 8257 


61 8362 


61 8466 


61 8571 


61 8676 


61 8780 


61 8884 


61 8989 


105 


416 


9093 


9198 


9302 


9406 


95 11 


9615 


9719 


9824 


9928 


62 0032 


104 


417 


620136 


62 0240 


620344 


62 0448 


620552 


62 0656 


620760 


62 0864 


62 0968 


1072 


104 


418 


1 1 76 


1280 


1384 


1488 


1592 


1695 


1799 


1903 


2007 


21 10 


104 


419 


2214 


2318 


2421 


2525 


2628 


2732 


2835 


2939 


3042 


3H6 


104 


420 


62 3249 


62 3353 


62 3456 


623559 


62 3663 


62 3766 


62 3869 


62 3973 


62 4076 


624179 


103 


421 


4282 


4385 


4488 


459i 


4695 


4798 


4901 


5004 


5107 


5210 


103 


422 


5312 


5415 


5518 


5621 


5724 


5827 


5929 


6032 


6135 


6238 


103 


423 


6340 


6443 


6546 


6648 


675 1 


6853 


6956 


7058 


7161 


7263 


103 


424 


7366 


7468 


757i 


7673 


7775 


7878 


7980 


8082 


8185 


8287 


102 


425 


62 8389 


62 8491 


62 8593 


62 8695 


628797 


62 8900 


62 9002 


62 9104 


62 9206 


62 9308 


102 


426 


9410 


95 12 


9613 


97i5 


9817 


9919 


63 0021 


630123 


63 0224 


630326 


102 


427 


63 0428 


63 0530 


630631 


630733 


63 0835 


63 0936 


1038 


"39 


1241 


1342 


102 


428 


1444 


1545 


1647 


1748 


1849 


I95 1 


2052 


2153 


2255 


2356 


101 


429 


2457 


2559 


2660 


2761 


2862 


2963 


3064 


3165 


3266 


3367 


IOI 


430 


63 3468 


63 3569 


63 3670 


633771 


63 3872 


63 3973 


63 4074 


63 4175 


63 4276 


63 4376 


IOI 


431 


4477 


4578 


4679 


4779 


4880 


4981 


5081 


5182 


5283 


5383 


IOI 


432 


5484 


5584 


5685 


5785 


5886 


5986 


6087 


6187 


6287 


6388 


IOO 


433 


6488 


6588 


6688 


6789 


6889 


6989 


70S9 


7189 


7290 


7390 


IOO 


434 


7490 


759o 


7690 


7790 


7890 


7990 


8090 


8190 


8290 


8389 


IOO 


435 


63 8489 


63 8589 


63 8689 


63 8789 


63 8888 


63 8988 


63 9088 


639188 


63 9287 


63 9387 


IOO 


436 


9486 


9586 


9686 


9785 


9885 


9984^640084 640183 


64 0283 


64 0382 


99 


437 


640481 


640581 


64 0680 


640779 


64 0879 


640978 1077; 1 1 77 


1276 


1375 


99 


438 


1474 


1573 


1672 


1771 


1871 


1970 


2069 


2168 


2267 


2366 


99 


439 


2465 


2563 


2662 


2761 


2860 


2959 


3058 


3156 


3255 


3354 


99 


440 


64 3453 


64 355 1 


64 3650 


64 3749 


64 3847 


64 3946 


64 4044 


644143 


64 4242 


64 4340 


98 


441 


4439 


4537 


4636 


4734 


4832 


493i 


5029 


5 I2 7 


5226 


5324 


98 


442 


5422 


552i 


5619 


5717 


5815 


5913 


601 1 


6110 


6208 


6306 


98 


443 


6404 


6502 


6600 


6698 


6796 


6894 6992 


7089 


7187 


7285 


98 


444 


7383 


748i 


7579 


7676 


7774 


7872 7969 


8067 


8165 


8262 


98 


445 


64 8360 


64 8458 


648555 


64 8653 


64 8750 


64 8848 64 8945 


64 9043 


649140 


64 9237 


97 


446 


9335 


9432 


953° 


9627 


9724 


9821 i 9919 


65 0016 


650113 


65 0210 


97 


447 


65 0308 


65 0405 


65 0502 


65 0599 


65 0696 


65 °793 65 0890 


0987 


1084 


1181 


97 


448 


1278 


1375 


1472 


1569 


1666 


1762 1859 1956 


2053 


2150 


97 


449 


2246 


2343 


2440 


2536 


2633 


2730 


2826 


2923 


3019 


3116 


97 


450 


653213 


65 3309 


65 3405 


65 3502 


65 3598 


65 3695 


65 379i 


65 3888 


65 3984 


65 4080 


96 


45i 


4177 


4273 


4369 


4465 


4562 


4658 


4754 


4850 


4946 


5042 


96 


452 


5138 


5 2 35 


533i 


5427 


5523 


5619 


5715 


5810 


5906 


6002 


96 


453 


6098 


6194 


6290 


6386 


6482 


6577 


6673 


6769 


6864 


6960 


96 


454 


7056 


7152 


7247 


7343 


7438 


7534 


7629 


7725 


7820 


7916 


96 


455 


65 801 1 


65 8107 


65 8202 


65 8298 


65 8393 


65 8488 


65 8584 


65 8679 


65 8774 


65 8870 


95 


456 


8965 


9060 


9155 


9250 


9346 


9441 


9536 


9631 


9726 


9821 


95 


457 


9916 


66 001 1 


66 0106 


660201 


660296 


660391 


66 0486 


660581 


66 0676 


660771 


95 


458 


66 0865 


0960 


io55 


1150 


1245 


1339 


1434 


1529 


1623 


1718 


95 


459 


1813 


1907 


2002 


2096 


2191 


2286 


2380 


2475 


2569 


2663 


95 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 



170 



LOGARITHMS OF NUMBERS. 



N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 

94 


460 


66 2758 


66 2852 


66 2947 


66 3041 


663135 


66 3230 


66 3324 


663418 


663512 


66 3607 


461 


37oi 


3795 


3889 


3983 


4078 


4172 


4266 


4360 


4454 


4548 


94 


462 


4642 


4736 


4830 


4924 


5018 


5112 


5206 


5299 


5393 


5487 


94 


463 


558i 


5675 


5769 


5862 


5956 


6050 


6143 


6237 


6331 


6424 


94 


464 


6518 


6612 


6705 


6799 


6892 


6986 


7079 


7173 


7266 


736o 


94 


465 


66 7453 


66 7546 


66 7640 


66 7733 


66 7826 


66 7920 


668013 


668106 


66 8199 


66 8293 


93 


466 


8386 


8479 


8572 


8665 


8759 


8852 


8945 


9038 


9131 


9224 


93 


467 


9317 


9410 


95°3 


9596 


9689 


9782 


9875 


9967 


67 0060 


670153 


93 


468 


67 0246 


67 0339 


670431 


670524 


670617 


670710 


67 0802 


67 0895 


0988 


1080 


93 


469 


"73 


1265 


1358 


145 1 


1543 


1636 


1728 


1821 


1913 


2005 


93 


470 


67 2098 


67 2190 


67 2283 


67 2375 


67 2467 


67 2560 


67 2652 


67 2744 


67 2836 


67 2929 


92 


471 


3021 


3H3 


3205 


3297 


339o 


3482 


3574 


3666 


3758 


3850 


92 


472 


3942 


4034 


4126 


4218 


43io 


4402 


4494 


4586 


4677 


4769 


92 


473 


4861 


4953 


5°45 


5137 


5228 


5320 


5412 


5503 


5595 


5687 


92 


474 


5778 


5870 


5962 


6053 


6145 


6236 


6328 


6419 


6511 


6602 


92 


475 


67 6694 


67 6785 


67 6876 


67 6968 


67 7059 


677151 


67 7242 


67 7333 


67 7424 


67 7516 


9i 


476 


7607 


7698 


7789 


7881 


7972 


8063 


8i54 


8245 


8336 


8427 


9i 


477 


8518 


8609 


8700 


8791 


8882 


8973 


9064 


9155 


9246 


9337 


91 


478 


9428 


95*9 


9610 


9700 


9791 


9882 


9973 


68 0063 


680154 


68 0245 


91 


479 


680336 


68 0426 


680517 


68 0607 


68 0698 


680789 


680879 


0970 


1060 


"5 1 


91 


480 


68 1 241 


68 1332 


68 1422 


681513 


68 1603 


68 1693 


681784 


68 1874 


68 1964 


68 2055 


90 


481 


2145 


2235 


2326 


2416 


2506 


2596 


2686 


2777 


2867 


2957 


90 


482 


3047 


3137 


3227 


3317 


3407 


3497 


3587 


3677 


3767 


3857 


90 


483 


3947 


4037 


4127 


4217 


4307 


4396 


4486 


4576 


4666 


4756 


90 


484 


4845 


4935 


5° 2 5 


5 JI 4 


5204 


5294 


5383 


5473 


5563 


5652 


90 


485 


68 5742 


68 5831 


685921 


68 6010 


68 6100 


686189 


68 6279 


68 6368 


68 6458 


68 6547 


89 


486 


6636 


6726 


6815 


6904 


6994 


7083 


7172 


7261 


735 * 


7440 


89 


487 


7529 


7618 


7707 


7796 


7886 


7975 


8064 


8i53 


8242 


833i 


89 


488 


8420 


8509 


8598 


8687 


8776 


8865 


8953 


9042 


9131 


9220 


89 


489 


9309 


9398 


9486 


9575 


9664 


9753 


9841 


9930 


690019 


690107 


89 


490 


690196 


69 0285 


69 0373 


69 0462 


690550 


69 0639 


69 0728 


690816 


69 0905 


69 0993 


89 


491 


1081 


1 1 70 


1258 


1347 


1435 


1524 


1612 


1700 


1789 


1877 


88 


492 


1965 


2053 


2142 


2230 


2318 


2406 


2494 


2583 


2671 


2759 


88 


493 


2847 


2935 


3023 


3111 


3199 


- 3287 


3375 


3463 


355i 


3639 


88 


494 


3727 


3815 


3903 


399i 


4078 


4166 


4254 


4342 


443° 


4517 


88 


495 


69 4605 


69 4693 


69 4781 


69 4868 


69 4956 


69 5°44 


695131 


695219 


69 5307 


69 5394 


88 


496 


5482 


5569 


5 6 57 


5744 


5832 


5919 


6007 


6094 


6182 


6269 


87 


497 


6356 


6444 


6531 


6618 


6706 


6793 


6880 


6968 


7055 


7142 


87 


498 


7229 


73i7 


7404 


7491 


7578 


7665 


7752 


7839 


7926 


8014 


87 


499 


8101 


8188 


8275 


8362 


8449 


8535 


8622 


8709 


8796 


8883 


87 


500 


69 8970 


699057 


699144 


699231 


699317 


69 9404 


69 9491 


699578 


69 9664 


699751 


87 


501 


9838 


9924 


70 001 1 


70 0098 


700184 


700271 


700358 


70 0444 


700531 


70 061 7 


87 


502 


70 0704 


70 0790 


0877 


0963 


1050 


1136 


1222 


1309 


1395 


1482 


86 


503 


1568 


1654 


1 741 


1827 


1913 


1999 


2086 


2172 


2258 


2344 


86 


504 


2431 


2517 


2603 


2689 


2775 


2861 


2947 


3033 


3ii9 


3205 


86 


505 


70 3291 


70 3377 


7° 3463 


7° 3549 


70 3635 


70 3721 


70 3807 


7° 3893 


7° 3979 


70 4065 


86 


506 


4I5 1 


4236 


4322 


4408 


4494 


4579 


4665 


475 1 


4837 


4922 


86 


507 


5008 


5°94 


5*79 


5265 


535° 


5436 


5522 


5607 


5693 


5778 


86 


508 


5864 


5949 


6035 


6120 


6206 


6291 


6376 


6462 


6547 


6632 


f 5 


509 


6718 


6803 


6888 


6974 


7059 


7144 


7229 


7315 


7400 


7485 


85 


5io 


70 7570 


70 7655 


70 7740 


70 7826 


70 791 1 


70 7996 


70 8081 


708166 


708251 


70 8336 


85 


5" 


8421 


8506 


8591 


8676 


8761 


8846 


8931 


9015 


9100 


9185 


f 5 


512 


9270 


9355 


9440 


. 9524 


9609 


9694 


9779 


9863 


9948 


71 0033 


85 


5i3 


710117 


71 0202 


71 0287 


71 0371 


71 0456 


71 0540 


71 0625 


71 0710 


71 0794 


0879 


85 


5i4 


0963 


1048 


1132 


1217 


1 301 


1385 


1470 


1554 


1639 


1723 


84 


515 


71 1807 


71 1892 


71 1976 


71 2060 


71 2144 


71 2229 


7 1 2313 


71 2397 


71 2481 


71 2566 


84 


5i6 


2650 


2734 


2818 


2902 


2986 


3070 


3i54 


3238 


3323 


3407 


84 


5i7 


3491 


3575 


3659 


3742 


3826 


3910 


3994 


4078 


4162 


4246 


84 


5i8 


4330 


4414 


4497 


4581 


4665 


4749 


4833 


4916 


5000 


5084 


84 


5i9 


5167 


5251 


5335 


5418 


55°2 


5586 


5669 


5753 


5836 


5920 


84 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 



LOGARITHMS OF NUMBERS. 



171 



N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 


520 


71 6003 


71 6087 


71 6170 


71 6254 


7 1 6 337 


71 6421 


71 6504 


71 6588 


71 6671 


71 6754 


^3 


521 


6838 


6921 


7004 


7088 


7171 


7254 


7338 


7421 


7504 


7587 


83 


522 


7671 


7754 


7837 


7920 


8003 


8086 


8169 


8253 


8336 


8419 


83 


523 


8502 


8585 


8668 


8751 


8834 


8917 


9000 


9083 


9165 


9248 


83 


524 


933i 


9414 


9497 


9580 


9663 


9745 


9828 


991 1 


9994 


720077 


83 


525 


720159 


72 0242 


72 0325 


72 0407 


72 0490 


720573 


720655 


720738 


720821 


72 0903 


83 


526 


0986 


1068 


"51 


1233 


1316 


1398 


1481 


1563 


1646 


1728 


82 


527 


1811 


1893 


1975 


2058 


2140 


2222 


2305 


2387 


2469 


255 2 


82 


528 


2634 


2716 


2798 


2881 


2963 


3045 


3 I2 7 


3209 


3291 


3374 


82 


529 


345 6 


3538 


3620 


3702 


3784 


3866 


3948 


4030 


4112 


4194 


82 


530 


724276 


72 4358 


72 4440 


72 4522 


72 4604 


72 4685 


724767 


72 4849 


72 4931 


72 5 OI 3 


82 


53i 


5°95 


5 J 7 6 


5258 


534o 


5422 


5503 


5585 


5667 


5748 


5830 


82 


532 


5912 


5993 


6075 


6156 


6238 


6320 


6401 


6483 


6564 


6646 


82 


533 


6727 


6809 


6890 


6972 


7053 


7134 


7216 


7297 


7379 


7460 


81 


534 


754i 


7623 


7704 


7785 


7866 


7948 


8029 


8110 


8191 


8273 


81 


535 


72 8354 


72 8435 


728516 


72 8597 


72 8678 


728759 


72 8841 


72 8922 


72 9003 


72 9084 


81 


536 


9165 


9246 


9327 


9408 


9489 


957° 


9651 


9732 


9813 


9893 


81 


537 


9974 


73 0055 


730136 


730217 


73 0298 


73 0378 


73 0459 


73 0540 


730621 


730702 


81 


538 


730782 


0863 


0944 


1024 


1 105 


1 186 


1266 


1347 


1428 


1508 


81 


539 


1589 


1669 


1750 


1830 


1911 


1 991 


2072 


2152 


2233 


2313 


81 


54o 


73 2394 


73 2474 


73 2555 


73 2635 


732715 


73 2796 


73 2876 


73 2956 


73 3037 


733H7 


80 


54i 


3197 


3278 


3358 


3438 


35i8 


3598 


3679 


3759 


3839 


39i9 


80 


542 


3999 


4079 


4160 


4240 


4320 


4400 


4480 


4560 


4640 


4720 


80 


543 


4800 


4880 


4960 


5040 


5120 


5200 


5 2 79 


5359 


5439 


55i9 


80 


544 


5599 


5 6 79 


5759 


5838 


59i8 


5998 


6078 


6i57 


6237 


6317 


80 


545 


73 6397 


73 647 6 


736556 


73 6635 


736715 


73 6795 


73 6874 


73 6954 


73 7034 


737H3 


80 


546 


7193 


7272 


7352 


743i 


75" 


759o 


7670 


7749 


7829 


7908 


79 


547 


7987 


8067 


8146 


8225 


8305 


8384 


8463 


8543 


8622 


8701 


79 


548 


8781 


8860 


8939 


9018 


9097 


9177 


9256 


9335 


9414 


9493 


79 


549 


9572 


9651 


973i 


9810 


9889 


9968 


740047 


740126 


74 0205 


74 0284 


79 


55o 


74 0363 


74 0442 


740521 


74 0600 


74 0678 


74 0757 


74 0836 


74 0915 


74 0994 


74 1073 


79 


55i 


1152 


1230 


1309 


1388 


1467 


1546 


1624 


1703 


1782 


i860 


79 


552 


1939 


2018 


2096 


2175 


2254 


2332 


241 1 


2489 


2568 


2647 


79 


553 


2725 


2804 


2882 


2961 


3039 


3118 


3196 


3275 


3353 


343i 


78 


554 


35 IO 


3588 


3667 


3745 


3823 


3902 


3980 


4058 


4136 


4215 


78 


555 


74 4293 


74 4371 


74 4449 


744528 


74 4606 


74 4684 


744762 


74 4840 


744919 


74 4997 


78 


556 


5°75 


5*53 


5231 


5309 


5387 


5465 


5543 


5621 


5699 


5777 


78 


557 


5855 


5933 


601 1 


6089 


6167 


6245 


6323 


6401 


6479 


6556 


78 


558 


6634 


6712 


6790 


6868 


6945 


7023 


7101 


7179 


7256 


7334 


78 


559 


7412 


7489 


7567 


7645 


7722 


7800 


7878 


7955 


8033 


8110 


78 


560 


748188 


74 8266 


74 8343 


74 8421 


74 8498 


748576 


74 8653 


748731 


74 8808 


74 8885 


77 


56i 


8963 


9040 


9118 


9195 


9272 


935° 


9427 


9504 


9582 


9659 


77 


562 


9736 


9814 


9891 


9968 


75 0045 


750123 


75 0200 


750277 


75 0354 


75 0431 


77 


563 


75 °5o8 


75 0586 


75 o66 3 


75 0740 


0817 


0894 


0971 


1048 


1125 


1202 


77 


564 


1279 


1356 


H33 


1510 


1587 


1664 


1 741 


1818 


1895 


1972 


77 


565 


75 2048 


752125 


75 2202 


75 2279 


75 2356 


75 2433 


75 2509 


75 2586 


75 2663 


75 2740 


77 


566 


2816 2893 


2970 


3047 


3123 


3200 


3277 


3353 


343° 


3506 


77 


567 


3583 


3660 


3736 


3813 


3889 


3966 


4042 


4119 


4195 


4272 


77 


568 


4348 


4425 


45 QI 


4578 


4654 


473o 


4807 


4883 


4960 


5036 


76 


569 


5112 


5189 


5265 


534i 


54i7 


5494 


557o 


5646 


5722 


5799 


76 


57o 


75 5875 


75 5951 


75 6027 


75 6l °3 


75 6180 


75 6256 


75 6332 


75 6408 


75 6484 


75 6560 


76 


57i 


6636 


6712 


6788 


6864 


6940 


7016 


7092 


7168 


7244 


7320 


76 


572 


7396 


7472 


7548 


7624 


7700 


7775 


7851 


7927 


8003 


8079 


76 


573 


8i55 


8230 


8306 


8382 


8458 


8533 


8609 


8685 


8761 


8836 


76 


574 


8912 


8988 


9063 


9i39 


9214 


9290 


9366 


9441 


9517 


9592 


76 


575 


75 9668 


75 9743 


759819 


75 9894 


75 997o 


76 0045 


760121 


760196 


760272 


760347 


75 


576 


76 0422 


76 0498 


760573 


76 0649 


76 0724 


0799 


0875 


0950 


1025 


IIOI 


75 


577 


1 1 76 


1251 


1326 


1402 


1477 


1552 


1627 


1702 


1778 


1853 


75 


578 


1928 


2003 


2078 


2153 


2228 


2303 


2378 


2453 


2529 


2604 


75 


579 


2679 


2754 


2829 


2904 


2978 


3053 


3128 


3203 


3278 


3353 


75 


N. 





1 


2 


3 


4 ' 


5 


6 


7 


8 


9 D. 



172 



LOGARITHMS OF NUMBERS. 



N. 


O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 

75 


580 


76 3428 ! 76 3503 76 3578 1 76 3653 


763727 


76 3802 


763877 


76 3952 1 76 4027 


764101 


581 


4176 


4251 


4326 4400 


4475 


455o 


4624 


4699 


4774 


4848 


75 


582 


4923 


4998 


5°72 5H7 


5221 


5296 


537o 


5445 


552o 


5594 


75 


583 


5669 


5743 


5818 5892 


5966 


6041 


6115 


6190 


6264 


6338 


74 


584 


6413 


6487 


6562 


6636 


6710 


6785 


6859 


6933 


7007 


7082 


74 


585 


767156 


76 7230 


76 7304 


7 6 7379 


76 7453 


76 7527 


76 7601 


767675 767749 


76 7823 


74 


586 


7898 


7972 


8046 


8120 


8194 


8268 


8342 


8416 


8490 


8564 


74 


587 


8638 


8712 


8786 


8860 


8934 


9008 


9082 


9i5 6 


9230 


9303 


74 


588 


9377 


945i 


9525 


9599 


9673 


9746 


9820 


9894 


9968 


77 0042 


74 


589 


770115 


77 0189 


77 0263 


77 °336 


77 0410 


77 0484 


77 0557 


770631 


770705 


0778 


74 


590 


770852 


77 0926 


77 °999 


77 io 73 


77 1 146 


77 1220 


77 1293 


77 1367 


77 1440 


77I5H 


74 


59i 


1587 


1661 


1734 


1808 


1881 


1955 


2028 


2102 


2175 


2248 


73 


592 


2322 


2395 


2468 


2542 


2615 


2688 


2762 


2835 


2908 


2981 


73 


593 


3055 


3128 


3201 


3274 


3348 


342i 


3494 


35 6 7 


3640 


37*3 


73 


594 


3786 


3860 


3933 


4006 


4079 


4152 


4225 


4298 


437i 


4444 


73 


595 


77 4517 


77 4590 


77 4663 


77 4736 


77 4809 


77 4882 


77 4955 


77 5028 


775100 


77 5173 


73 


596 


5246 


53i9 


5392 


5465 


5538 


5610 


5683 


5756 


5829 


5902 


73 


597 


5974 


6047 


6120 


6193 


6265 


6338 


641 1 


6483 


6556 


6629 


73 


598 


6701 


6774 


6846 


6919 


6992 


7064 


7137 


7209 


7282 


7354 


73 


599 


7427 


7499 


7572 


7 6 44 


7717 


7789 


7862 


7934 


8006 


8079 


72 


600 


778151 


77 8224 


77 8296 


77 8368 


77 8441 


778513 


778585 


77 8658 


77 8730 


77 8802 


72 


601 


8874 


8947 


9019 


9091 


9163 


9236 


93o8 


9380 


9452 


9524 


72 


602 


9596 


9669 


9741 


9813 


9885 


9957 


78 0029 


780101 


780173 


780245 


72 


603 


780317 


78 0389 


78 0461 


780533 


78 0605 


78 0677 


0749 


0821 


0893 


0965 


72 


604 


io37 


1 109 


1181 


1253 


1324 


1396 


1468 


1540 


1612 


1684 


72 


605 


78 1755 


78 1827 


78 1899 


78 1971 


78 2042 


78 2114 


782186 


78 2258 


78 2329 


78 2401 


72 


606 


2473 


2544 


2616 


2688 


2759 


2831 


2902 


2974 


3046 


3H7 


72 


607 


3189 


3260 


3332 


3403 


3475 


3546 


3618 


3689 


376i 


3832 


7i 


608 


3904 


3975 


4046 


4118 


4189 


4261 


4332 


4403 


4475 


4546 


7i 


609 


4617 


4689 


4760 


4831 


4902 


4974 


5045 


5116 


5187 


5259 


7i 


610 


78 533o 


78 5401 


78 5472 


78 5543 


785615 


78 5686 


78 5757 


78 5828 


78 5899 


78 597° 


7i 


611 


6041 


6112 


6183 


6254 


6325 


6396 


6467 


6538 


6609 


6680 


7i 


612 


675 1 


6822 


6S93 


6964 


7035 


7106 


7177 


7248 


7319 


739o 


7i 


613 


7460 


753i 


7602 


7673 


7744- 


7815 


7885 


795 6 


8027 


8098 


7i 


614 


8168 


8239 


8310 


8381 


8451 


8522 


8593 


8663 


8734 


8804 


7i 


6i5 


788875 


78 8946 


78 9016 


78 9087 


789157 


789228 


78 9299 


78 9369 


78 9440 


789510 


7i 


616 


9581 


9651 


9722 


9792 


9863 


9933 


79 0004 


79 0074 


790144 


790215 


70 


617 


79 0285 


790356 


79 0426 


79 °49 6 


79 0567 


790637 


0707 


0778 


0848 


0918 


70 


5i8 


0988 


1059 


1 1 29 


1199 


1269 


1340 


1410 


1480 


i55o 


1620 


70 


619 


1691 


1761 


1831 


1901 


1971 


2041 


2111 


2181 


2252 


2322 


70 


620 


79 2392 


79 2462 


79 2532 


79 2602 


79 2672 


79 2742 


792812 


79 2882 


792952 


79 3022 


7° 


621 


3092 


3162 


3231 


33oi 


337i 


344i 


35" 


358i 


3651 


372i 


70 


622 


379o 


3860 


393o 


4000 


4070 


4139 


4209 


4279 


4349 


4418 


7° 


623 


4488 


4558 


4627 


4697 


4767 


4836 


4906 


4976 


5°45 


5115 


7° 


624 


5185 


5254 


5324 


5393 


5463 


5532 


5602 


5672 


574i 


5811 


70 


625 


79 5880 


79 5949 


79 6019 


79 6088 


796158 


79 6227 


79 6297 


79 6366 


79 6436 


79 6505 


69 


626 


6574 


6644 


6713 


6782 


6852 


6921 


6990 


7060 


7129 


7198 


69 


627 


7268 


7337 


7406 


7475 


7545 


7614 


7683 


7752 


7821 


7890 


69 


628 


7960 


8029 


8098 


8167 


8236 


8305 


8374 


8443 


8513 


8582 


69 


629 


8651 


8720 


8789 


8858 


8927 


8996 


9065 


9134 


9203 


9272 


69 


630 


79 9341 


79 9409 


79 9478 


79 9547 


79 9616 


79 9685 


79 9754 


799823 


79 9892 


799961 


69 


631 


80 0029 


80 0098 


800167 


800236 


80 0305 


80 0373 


80 0442 


80 05 1 1 


So 0580 


80 0648 


69 


632 


0717 


0786 


0854 


0923 


0992 


1061 


1 1 29 


1198 


1266 


1335 


69 


633 


1404 


1472 


i54i 


1609 


1678 


1747 


1815 


1884 


I95 2 


2021 


69 


634 


2089 


2158 


2226 


2295 


2363 


2432 


2500 


2568 


2637 


2705 


68 


635 


80 2774 


80 2842 


80 2910 


80 2979 


80 3047 


80 31 16 


803184 


803252 


803321 


80 3389 


68 


636 


3457 


3525 


3594 


3662 


373o 


3798 


3867 


3935 


4003 


4071 


68 


537 


4139 


4208 


4276 


4344 


4412 


4480 


4548 


4616 


4685 


4753 


68 


538 


4821 


4889 


4957 


5° 2 5 


5°93 


5161 


5229 


5297 


5365 


5433 


68 


639 

N. 


55 01 


5569 


5637 


5705 


5773 


5841 


59o8 


5976 


6044 


6112 


68 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 



LOGARITHMS OF NUMBERS. 



N. 





1. 


2 


3 


4 


5 


6 


7 


8 


9 


D. 


640 


806180 


80 6248 


806316 


80 6384 


80 645 1 


806519 


80 6587 


80 6655 


80 6723 


80 6790 


68 


641 


6858 


6926 


6994 


7061 


7129 


7197 


7264 


7332 


7400 


7467 


68 


642 


7535 


7603 


7670 


7738 


7806 


7873 


7941 


8008 


8076 


8143 


68 


643 


821 1 


8279 


8346 


8414 


8481 


8549 


8616 


8684 


8751 


8818 


67 


644 


8886 


8953 


9021 


9088 


915 6 


9223 


9290 


9358 


9425 


9492 


67 


645 


80 9560 


80 9627 


80 9694 


80 9762 


80 9829 


80 9896 


80 9964 


81 0031 


81 0098 


81 0165 


67 


646 


81 0233 


81 0300 


81 0367 


81 0434 


81 0501 


81 0569 


81 0636 


0703 


0770 


0837 


67 


647 


0904 


0971 


1039 


1 106 


1173 


1240 


1307 


1374 


1 441 


1508 


67 


648 


1575 


1642 


1709 


1776 


1843 


1910 


1977 


2044 


2111 


2178 


67 


649 


2245 


2312 


2379 


2445 


2512 


2579 


2646 


2713 


2780 


2847 


67 


650 


81 2913 


81 2980 


81 3047 


81 3114 


81 3181 


81 3247 


81 33H 


81 3381 


81 3448 


81 35H 


67 


651 


358i 


3648 


37H 


3781 


3848 


39i4 


3981 


4048 


4114 


4181 


67 


652 


4248 


4314 


4381 


4447 


45 J 4 


4581 


4647 


47*4 


4780 


4847 


67 


653 


4913 


4980 


5046 


5"3 


5 X 79 


5246 


5312 


5378 


5445 


55 11 


66 


654 


5578 


5 6 44 


57" 


5777 


5843 


59io 


5976 


6042 


6109 


6i75 


66 


655 


81 6241 


81 6308 


81 6374 


81 6440 


81 6506 


81 6573 


81 6639 


81 6705 


81 6771 


81 6838 


66 


656 


6904 


6970 


7036 


7102 


7169 


7235 


73oi 


7367 


7433 


7499 


66 


657 


75 6 5 


7 6 3i 


7698 


7764 


7830 


7896 


7962 


8028 


8094 


8160 


66 


658 


8226 


8292 


8358 


8424 


8490 


8556 


8622 


8688 


8754 


8820 


66 


659 


8885 


8951 


9017 


9083 


9149 


9215 


9281 


9346 


9412 


9478 


66 


660 


81 9544 


81 9610 


81 9676 


81 9741 


81 9807 


81 9873 


81 9939 


82 0004 


820070 


820136 


66 


661 


82 0201 


820267 


82 0333 


82 0399 


82 0464 


820530 


820595 


0661 


0727 


0792 


66 


662 


0858 


0924 


0989 


io 55 


1 1 20 


1186 


1251 


1317 


1382 


1448 


66 


663 


i5H 


1579 


1645 


1710 


1775 


1841 


1906 


1972 


2037 


2103 


65 


664 


2168 


2233 


2299 


2364 


2430 


2495 


2560 


2626 


2691 


2756 


65 


665 


82 2822 


82 2887 


82 2952 


823018 


82 3083 


82 3148 


823213 


82 3279 


82 3344 


82 3409 


65 


666 


3474 


3539 


3605 


3670 


3735 


3800 


3865 


393o 


3996 


4061 


65 


667 


4126 


4191 


4256 


4321 


4386 


445 1 


45 l6 


4581 


4646 


4711 


65 


668 


4776 


4841 


4906 


4971 


5°36 


5101 


5166 


5231 


5296 


536i 


65 


669 


5426 


549i 


5556 


5621 


5686 


575 1 


5815 


5880 


5945 


6010 


65 


670 


82 6075 


82 6140 


82 6204 


82 6269 


82 6334 


82 6399 


82 6464 


826528 


82 6593 


82 6658 


65 


671 


6723 


6787 


6852 


6917 


6981 


7046 


71 1 1 


7175 


7240 


7305 


65 


672 


7369 


7434 


7499 


7563 


7628 


7692 


7757 


7821 


7886 


795 1 


65 


673 


8015 


8080 


8144 


8209 


8273 


8338 


8402 


8467 


8531 


8595 


64 


674 


8660 


8724 


8789 


8853 


8918 


8982 


9046 


9111 


9175 


9239 


64 


675 


82 9304 


82 9368 


82 9432 


82 9497 


829561 


82 9625 


82 9690 


82 9754 


829818 


82 9882 


64 


676 


9947 


83 001 1 


^3 °°75 


830139 


83 0204 


83 0268 


830332 


83 0396 


83 0460 


830525 


64 


677 


830589 


0653 


0717 


0781 


0845 


0909 


0973 


io37 


1 102 


1 166 


64 


678 


1230 


1294 


1358 


1422 


i486 


1550 


1614 


1678 


1742 


1806 


64 


679 


1870 


1934 


1998 


2062 


2126 


2189 


2253 


2317 


2381 


2445 


64 


680 


83 2509 


83 2573 


83 2637 


83 2700 


83 2764 


83 2828 


83 2892 


832956 


83 3°2o 


833083 


64 


681 


, 3H7 


3211 


3275 


3338 


3402 


3466 


353o 


3593 


3657 


372i 


64 


682 


3784 


3848 


3912 


3975 


4039 


4103 


4166 


4230 


4294 


4357 


64 


683 


4421 


4484 


4548 


461 1 


4675 


4739 


4802 


4866 


4929 


4993 


64 


684 


5056 


5120 


5183 


5247 


53io 


5373 


5437 


55°o 


5564 


5627 


63 


685 


83 5691 


83 5754 


835817 


835881 


^3 5944 


83 6007 


83 6071 


836134 


836197 


836261 


63 


686 


6324 


6387 


645 1 


65H 


6577 


6641 


6704 


6767 


6830 


6894 


63 


687 


6957 


7020 


7083 


7146 


7210 


7273 


7336 


7399 


7462 


7525 


63 


688 


7588 


7652 


7715 


7778 


7841 


7904 


7967 


8030 


8093 


8156 


63 


689 


8219 


8282 


8345 


8408 


8471 


8534 


8597 


8660 


8723 


8786 


63 


6go 


83 8849 


838912 


83 8975 


83 9038 


839101 


839164 


839227 


83 9289 


839352 


83 9415 


63 


691 


9478 


\ 954i 


9604 


9667 


9729 


9792 


9855 99i8 


9981 


84 0043 


63 


6g2 


840106 


840169 


840232 


84 0294 


840357 


84 0420 


84 0482 


840545 


84 0608 


0671 


63 


693 


0733 


0796 


0859 


0921 


0984 


1046 


1 109 


1 172 


1234 


1297 


63 


694 


1359 


1422 


1485 


1547 


1610 


1672 


1735 


1797 


i860 


1922 


63 


695 


84 1985 


84 2047 


84 21 10 


84 2172 


84 2235 


84 2297 


84 2360 


84 2422 84 2484 


84 2547 


62 


696 


2609 


2672 


2734 


2796 


2859 


2921 


2983 


3046 3108 


3170 


62 


697 


3233 


3295 


3357 


3420 


3482 


3544 


3606 


3669 


3731 


3793 


62 


698 


3855 


39i8 


398o 


4042 


4104 


4166 


4229 


4291 


4353 


4415 


62 


699 


4477 


4539 


4601 


4664 


4726 


4788 


4850 


4912 


4974 


5°36 


62 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 



174 



LOGARITHMS OF NUMBERS. 



N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 

62 


700 


84 5098 


845160 


84 5222 


84 5284 


84 5346 


84 5408 


84 5470 


84 5532 


84 5594 


84 5656 


701 


5718 


5780 


5842 


5904 


5966 


6028 


6090 


6151 


6213 


6275 


62 


702 


6337 


6399 


6461 


6523 


6585 


6646 


6708 


6770 


6832 


6894 


62 


703 


6955 


7017 


7079 


7141 


7202 


7264 


7326 


7388 


7449 


75 11 


62 


704 


7573 


7 6 34 


7696 


7758 


7819 


7881 


7943 


8004 


8066 


8128 


62 


705 


848189 


848251 


848312 


848374 


848435 


84 8497 


848559 


84 8620 


84 8682 


848743 


62 


706 


8805 


8866 


8928 


8989 


9051 


9112 


9174 


9235 


9297 


9358 


61 


707 


9419 


948i 


9542 


9604 


9665 


9726 


9788 


9849 


991 1 


9972 


61 


708 


85 0033 


85 0095 


850156 


850217 


85 0279 


85 0340 


85 0401 


85 0462 


85 0524 


850585 


61 


709 


0646 


0707 


0769 


0830 


0891 


0952 


1014 


1075 


1136 


"97 


61 


710 


85 1258 


85 1320 


85 1381 


851442 


85 1503 


85 1564 


85 1625 


85 1686 


85 1747 


85 1809 


61 


711 


1870 


i93i 


1992 


2053 


2114 


2175 


2236 


2297 


2358 


2419 


61 


712 


2480 


2541 


2602 


2663 


2724 


2785 


2846 


2907 


2968 


3029 


61 


713 


3090 


315° 


321 1 


3272 


3333 


3394 


3455 


35 l6 


3577 


3637 


61 


714 


3698 


3759 


3820 


3881 


394 1 


4002 


4063 


4124 


4185 


4245 


61 


715 


85 43°6 


85 4367 


85 4428 


85 4488 


85 4549 


85 4610 


85 4670 


85 473i 


85 4792 


854852 


61 


716 


4913 


4974 


5°34 


5°95 


5 J 5 6 


5216 


5277 


5337 


5398 


5459 


61 


717 


55 J 9 


558o 


5640 


57oi 


576i 


5822 


5882 


5943 


6003 


6064 


61 


718 


6124 


6185 


6245 


6306 


6366 


6427 


6487 


6548 


6608 


6668 


60 


719 


6729 


6789 


6850 


6910 


6970 


7°3i 


7091 7152 


7212 


7272 


60 


720 


85 7332 


85 7393 


85 7453 


85 7513 


85 7574 


85 7634 


85 7694 


85 7755 


85 7815 


85 7875 


60 


721 


7935 


7995 


8056 


8116 


8176 


8236 


8297 


8357 


8417 


8477 


60 


722 


8537 


8597 


8657 


8718 


8778 


8838 


8898 


8958 


9018 


9078 


60 


723 


9138 


9198 


9258 


93i8 


9379 


9439 


9499 


9559 


9619 


9679 


60 


724 


9739 


9799 


9859 


9918 


9978 


86 0038 


86 0098 


860158 


860218 


86 0278 


60 


725 


860338 


86 0398 


86 0458 


860518 


860578 


86 0637 


86 0697 


860757 


860817 


860877 


60 


726 


0937 


0996 


1056 


1116 


1176 


1236 


1295 


1355 


1415 


1475 


60 


727 


1534 


1594 


1654 


1714 


1773 


1833 


1893 


1952 


2012 


2072 


60 


728 


2131 


2191 


2251 


2310 


2370 


2430 


2489 


2549 


2608 


2668 


60 


729 


2728 


2787 


2847 


2906 


2966 


3025 


3085 


3H4 


3204 


3263 


60 


730 


86 3323 


86 3382 


86 3442 


86 3501 


863561 


86 3620 


86 3680 


86 3739 


86 3799 


86 3858 


59 


73i 


3917 


3977 


4036 


4096 


4155 


4214 


4274 


4333 


4392 


445 2 


59 


732 


45 11 


457o 


4630 


4689 


4748 


. 4808 


4867 


4926 


4985 


5°45 


59 


733 


5^4 


5 l6 3 


5222 


5282 


534i 


5400 


5459 


5519 


5578 


5 6 37 


59 


734 


5696 


5755 


5814 


5874 


5933 


5992 


6051 


6110 


6169 


6228 


59 


735 


86 6287 


86 6346 


86 6405 


86 6465 


86 6524 


86 6583 


86 6642 


86 6701 


86 6760 


866819 


59 


736 


6878 


6937 


6996 


7055 


7114 


7 J 73 


7232 


7 o2 J 


735o 


7409 


59 


737 


7467 


7526 


7585 


7644 


7703 


7762 


7821 


7880 


7939 


7998 


59 


738 


8056 


8115 


8174 


8233 


8292 


8350 


8409 


8468 


8527 


8586 


59 


739 


8644 


8703 


8762 


8821 


8879 


8938 


8997 


9056 


9114 


9173 


59 


740 


869232 


86 9290 


86 9349 


86 9408 


86 9466 


869525 


86 9584 


86 9642 


869701 


86 9760 


59 


74i 


9818 


9877 


9935 


9994 


870053 


87 OIII 


870170 


870228 


87 0287 


^7 °345 


59 


742 


87 0404 


87 0462 


870521 


870579 


0638 


0696 


°755 


0813 


0872 


0930 


58 


743 


0989 


1047 


1 106 


1 164 


1223 


1281 


1339 


1398 


I45 6 


1515 


5 * 


744 


1573 


1631 


1690 


1748 


1806 


1865 


1923 


1981 


2040 


2098 


58 


745 


872156 


872215 


87 2273 


87 2331 


87 2389 


872448 


87 2506 


87 2564 


87 2622 


87 2681 


5 ! 


746 


2739 


2797 


2855 


2913 


2972 


3°3° 


3088 


3146 


3204 


3262 


5 f 


747 


332i 


3379 


3437 


3495 


3553 


361 1 


3669 


3727 


3785 


3844 


58 


748 


3902 


3960 


4018 


4076 


4134 


4192 


4250 


4308 


4366 


4424 


58 


749 


4482 


4540 


4598 


4656 


4714 


4772 


4830 


4888 


4945 


5°°3 


58 


75o 


87 5061 


87 5"9 


875177 


87 5235 


875293 


87 5351 


87 5409 


87 5466 


87 5524 


87 5582 


58 


75i 


5640 


5698 


5756 


5813 


5871 


5929 


5987 


6045 


6102 


6160 


58 


752 


6218 


6276 


6333 


6391 


6449 


6507 


6564 


6622 


6680 


6737 


58 


753 


6795 


6853 


6910 


6968 


7026 


7083 


7141 


7199 


7256 


73H 


5 f 


754 


737 1 


7429 


7487 


7544 


7602 


7659 


7717 


7774 


7832 


7889 


58 


755 


87 7947 


87 8004 


87 8062 


878119 


878177 


87 8234 


87 8292 


87 8349 


87 8407 


87 8464 


57 


756 


8522 


8579 


8637 


8694 


8752 


8809 


8866 


8924 


8981 


9039 


57 


757 


9096 


9153 


921 1 


9268 


9325 


9383 


9440 


9497 


9555 


9612 


57 


758 


9669 


9726 


9784 


9841 


9898 


995 6 


880013 


88 0070 


880127 


880185 


57 


759 


88 0242 


88 0299 


88 0356 


880413 


880471 


88 0528 


0585 


0642 


6699 


0756 


57 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 



LOGARITHMS OF NUMBERS. 



175 



N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 


760 


880814 


880871 


88 0928 


88 0985 


88 1042 


88 1099 


881156 


881213 


88 1271 


88 1328 


57 


761 


1385 


1442 


1499 


1556 


1613 


1670 


1727 


1784 


1841 


1898 


57 


762 


1955 


2012 


2069 


2126 


2183 


2240 


2297 2354 


241 1 


2468 


57 


763 


25 2 5 


2581 


2638 


2695 


2752 


2809 


2866 


2923 


2980 


3037 


57 


764 


3°93 


315° 


3207 


3264 


332i 


3377 


3434 


3491 


3548 


3605 


57 


765 


88 3661 


883718 


88 3775 


88 3832 


88 3888 


88 3945 


88 4002 


88 4059 


884115 


884172 


57 


766 


4229 


4285 


4342 


4399 


4455 


45 12 


45 6 9 


4625 


4682 


4739 


57 


767 


4795 


4852 


4909 


4965 


5022 


5078 


5135 


5 J 92 


5248 


5305 


57 


768 


536i 


54i8 


5474 


5531 


5587 


5 6 44 


57oo 


5757 


5813 


5870 


57 


769 


5926 


5983 


6039 


6096 


6152 


6209 


6265 


6321 


6378 


6434 


56 


770 


88 6491 


886547 


88 6604 


88 6660 


886716 


886773 


88 6829 


88 6885 


88 6942 


88 6998 


56 


771 


7°54 


71 1 1 


7167 


7223 


7280 


7336 


7392 


7449 


7505 


75 61 


56 


772 


7617 


7674 


773o 


7786 


7842 


7898 


7955 


801 1 


8067 


8123 


56 


773 


8179 


8236 


8292 


8348 


8404 


8460 


8516 


8573 


8629 


8685 


56 


774 


8741 


8797 


8853 


8909 


8965 


9021 


9077 1 9134 


9190 


9246 


56 


775 


88 9302 


889358 


88 9414 


88 9470 


889526 


889582 


88 9638 


88 9694 


88 9750 


88 9806 


56 


776 


9862 


9918 


9974 


89 0030 


89 0086 


89 0141 


89 0197 


890253 


89 0309 


89 0365 


56 


777 


89 0421 


89 0477 


890533 


0589 


0645 


0700 


0756 


0812 


0868 


0924 


^ 


778 


0980 


1035 


1091 


1 147 


1203 


1259 


I3H 


1370 


1426 


1482 


56 


779 


1537 


1593 


1649 


!7°5 


1760 


1816 


1872 


1928 


1983 


2039 


56 


780 


89 2095 


89 2150 


89 2206 


89 2262 


892317 


89 2373 


89 2429 


89 2484 


89 2540 


89 2595 


56 


781 


2651 


2707 


2762 


2818 


2873 


2929 


2985 


3040 


3096 


3I5 1 


56 


782 


3207 


3262 


33i8 


3373 


3429 


3484 


354o 


3595 


3651 


3706 


56 


783 


3762 


3817 


3873 


3928 


3984 


4039 


4094 


4150 


4205 


4261 


55 


784 


43i6 


437 1 


4427 


4482 


4538 


4593 


4648 


4704 


4759 


4814 


55 


785 


89 4870 


89 4925 


89 4980 


89 5°36 


89 5091 


895146 


89 5201 


89 5257 


89 5312 


89 5367 


55 


786 


5423 


5478 


5533 


5588 


5 6 44 


5 6 99 


5754 


5809 


5864 


592o 


55 


787 


5975 


6030 


6085 


6140 


6i95 


6251 


6306 


6361 


6416 


6471 


55 


788 


6526 


6581 


6636 


6692 


6747 


6802 


6857 


6912 


6967 


7022 


55 


789 


7077 


7132 


7187 


7242 


7297 


7352 


7407 


7462 


75 x 7 


7572 


55 


790 


89 7627 


89 7682 


89 7737 


89 7792 


89 7847 


89 7902 


89 7957 


898012 


89 8067 


898122 


55 


791 


8176 


8231 


8286 


8341 


8396 


8451 


8506 


8561 


8615 


8670 


55 


792 


8725 


8780 


8835 


8890 


8944 


8999 


9054 


9109 


9164 


9218 


55 


793 


9273 


9328 


9383 


9437 


9492 


9547 


9602 


9656 


9711 


9766 


55 


794 


9821 


9875 


9930 


9985 


90 0039 


90 0094 


900149 


90 0203 


900258 


900312 


55 


795 


90 0367 


90 0422 


90 0476 


900531 


900586 


90 0640 


90 0695 


90 0749 


90 0804 


900859 


55 


796 


0913 


0968 


1022 


1077 


1131 


1186 


1240 


1295 


1349 


1404 


55 


797 


1458 


1513 


!5 6 7 


1622 


1676 


i73i 


1785 


1840 


1894 


1948 


54 


798 


2003 


2057 


2112 


2166 


2221 


2275 


2329 


2384 


2438 


2492 


54 


799 


2547 


2601 


2655 


2710 


2764 


2818 


2873 


2927 


2981 


3036 


54 


800 


90 3090 


903144 


903199 


90 3253 


90 3307 


90 3361 


90 34i6 


9° 347o 


90 3524 


90 3578 


54 


801 


3633 


3687 


374i 


3795 


3849 


3904 


3958 


4012 


4066 


4120 


54 


802 


4174 


4229 


4283 


4337 


4391 


4445 


4499 


4553 


4607 


4661 


54 


803 


4716 


4770 


4824 


4878 


4932 


4986 


5040 


5094 


5H8 


5202 


54 


804 


5256 


53 IQ 


5364 


54i8 


5472 


5526 


558o 


5 6 34 


5688 


5742 


54 


805 


90 5796 


90 5850 


90 5904 


90 5958 


90 6012 


90 6066 


90 61 19 


906173 


90 6227 


906281 


54 


806 


6335 


6389 


6443 


6497 


6 55* 


6604 


6658 


6712 


6766 


6820 


54 


807 


6S74 


6927 


6981 


7°35 


7089 


7*43 


7196 


7250 


7304 


7358 


54 


808 


741 1 


7465 


75*9 


7573 


7626 


7680 


7734 


7787 


7841 


7895 


54 


809 


7949 


8002 


8056 


8110 


8163 


8217 


8270 


8324 


8378 


8431 


54 


810 


90 8485 


90 8539 


90 8592 


90 8646 


90 8699 


908753 


90 8807 


90 8860 


90 8914 


90 8967 


54 


811 


9021 


9074 


9128 


9181 


9235 


9289 


9342 


9396 


9449 


9503 


54 


812 


955 6 


9610 


9663 


9716 


9770 


9823 


9877 


993° 


9984 


91 0037 


53 


813 


91 0091 


91 0144 


91 0197 


91 0251 


91 0304 


91 0358 


91 041 1 


91 0464 


91 0518 


0571 


53 


814 


0624 


0678 


0731 


0784 


0838 


0891 


0944 


0998 


105 1 


1 104 


53 


815 


911158 


91 1211 


91 1264 


9i 1317 


9i 1371 


91 1424 


91 1477 


91 1530 


91 1584 


91 1637 


53 


816 


1690 


1743 


1797 


1850 


1903 


1956 


2009 


2063 


2116 


2169 


53 


817 


2222 


2275 


2328 


2381 


2435 


2488 


2541 


2594 


2647 


2700 


53 


818 


2753 


2806 


2859 


2913 


2966 


3019 


3072 


3125 


3178 


3231 


53 


8ig 


3284 


3337 


339o 


3443 


3496 


3549 


3602 


3655 


37o8 


376i 


53 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 



176 



LOGARITHMS OF NUMBERS. 



N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 

53 


820 


91 3814 


91 3867 


91 3920 


9i 3973 


91 4026 


91 4079 


91 4132 


91 4184 


91 4237 


91 4290 


821 


4343 


4396 


4449 


4502 


4555 


4608 


4660 


47*3 


4766 


4819 


53 


822 


4872 


4925 


4977 


5°3o 


5083 


5 X 36 


5189 


5241 


5294 


5347 


53 


823 


5400 


5453 


5505 


5558 


5611 


5664 


57i6 


57 6 9 


5822 


5875 


53 


824 


5927 


598o 


6033 


6085 


6138 


6191 


6243 


6296 


6349 


6401 


53 


825 


91 6454 


91 6507 


9i 6559 


91 6612 


91 6664 


91 6717 


91 6770 


91 6822 


91 6875 


91 6927 


53 


826 


6980 


7033 


7085 


7138 


7190 


7243 


7295 


7348 


7400 


7453 


53 


827 


7506 


7558 


7611 


7663 


7716 


7768 


7820 


7873 


7925 


7978 


52 


828 


8030 


8083 


8i35 


8188 


8240 


8293 


8345 


8397 


8450 


8502 


52 


829 


8555 


8607 


8659 


8712 


8764 


8816 


8869 


8921 


8973 


9026 


52 


830 


91 9078 


91 9130 


91 9183 


9i 9235 


91 9287 


91 9340 


91 9392 


91 9444 


91 9496 


9i 9549 


52 


831 


9601 


9653 


9706 


9758 


9810 


9862 


9914 


9967 


920019 


920071 


52 


832 


920123 


920176 


920228 


92 0280 


920332 


92 0384 


92 0436 


92 0489 


0541 


0593 


52 


833 


0645 


0697 


0749 


0801 


0853 


0906 


0958 


IOIO 


1062 


1114 


52 


834 


1 166 


1218 


1270 


1322 


1374 


1426 


1478 


i53o 


1582 


1634 


52 


835 


92 1686 


921738 


92 1790 


92 1842 


92 1894 


92 1946 


92 1998 


92 2050 


92 2102 


922154 


52 


836 


2206 


2258 


2310 


2362 


2414 


2466 


2518 


2570 


2622 


2674 


52 


837 


2725 


2777 


2829 


2881 


2933 


2985 


3037 


3089 


3140 


3192 


52 


838 


3 2 44 


3296 


3348 


3399 


345i 


35°3 


3555 


3607 


3658 


37™ 


52 


839 


3762 


3814 


3865 


3917 


3969 


4021 


4072 


4124 


4176 


4228 


52 


840 


924279 


92 4331 


92 4383 


92 4434 


92 4486 


92 4538 


92 4589 


92 4641 


92 4693 


92 4744 


52 


841 


4796 


4848 


4899 


495 1 


5°°3 


5°54 


5106 


5 J 57 


5209 


5261 


52 


842 


5312 


5364 


5415 


5467 


5518 


5570 


5621 


5 6 73 


5725 


5776 


52 


843 


5828 


5879 


5931 


5982 


6034 


6085 


6137 


6188 


6240 


6291 


5i 


844 


6342 


6394 


6445 


6497 


6548 


6600 


6651 


6702 


6754 


6805 


5i 


845 


92 6857 


92 6908 


92 6959 


92 701 1 


92 7062 


927114 


927165 


92 7216 


92 7268 


927319 


5i 


846 


7370 


7422 


7473 


7524 


7576 


7627 


7678 


773o 


7781 


7832 


5i 


847 


7883 


7935 


7986 


8037 


8088 


8140 


8191 


8242 


8293 


8345 


5 1 


848 


8396 


8447 


8498 


8549 


8601 


8652 


8703 


8754 


8805 


8857 


5i 


849 


8908 


8959 


9010 


9061 


9112 


9163 


9215 


9266 


9317 


9368 


5 1 


850 


929419 


92 947° 


92 9521 


92 9572 


92 9623 


92 9674 


929725 


929776 


929827 


92 9879 


5 1 


851 


993° 


998i 


93 0032 


93 0083 


93oi34 


930185 


930236 


930287 


93 0338 


93 0389 


5 1 


852 


93 °44Q 


930491 


0542 


0592 


0643 


0694 


0745 


0796 


0847 


0898 


5 1 


853 


0949 


1000 


105 1 


1 102 


"53 


1204 


1254 


1305 


1356 


1407 


5 1 


854 


1458 


i5°9 


1560 


1610 


1661 


1712 


1763 


1814 


1865 


1915 


5 1 


855 


93 1966 


932017 


93 2068 


932118 


932169 


93 2220 


93 2271 


93 2322 


93 2372 


93 2423 


5i 


856 


2474. 


2524 


2575 


2626 


2677 


2727 


2778 


2829 


2879 


2930 


5 1 


857 


2981 


3031 


3082 


3133 


3183 


3234 


3285 


3335 


3386 


3437 


5 1 


858 


3487 


3538 


3589 


3639 


3690 


3740 


379i 


3841 


3892 


3943 


5i 


859 


3993 


4044 


4094 


4145 


4i95 


4246 


4296 


4347 


4397 


4448 


5 1 


860 


93 4498 


93 4549 


93 4599 


93 4650 


93 47oo 


93 475 1 


93 4801 


93 4852 


93 4902 


93 4953 


5° 


861 


5°°3 


5054 


5 io 4 


5*54 


5205 


5255 


53o6 


535 6 


5406 


5457 


5° 


862 


5507 


5558 


5608 


5658 


5709 


5759 


5809 


5860 


59io 


5960 


5° 


863 


601 1 


6061 


6111 


6162 


6212 


6262 


6313 


6363 


6413 


6463 


50 


864 


6514 


6564 


6614 


6665 


6 7 J 5 


6765 


6815 


6865 


6916 


6966 


5° 


865 


93 7° l6 


93 7066 


93 7U7 


937167 


937217 


93 7 26 7 


93 7317 


93 7367 


93 74i8 


93 7468 


50 


866 


7518 


7568 


7618 


7668 


7718 


7769 


7819 


7869 


7919 


7969 


5° 


867 


8019 


8069 


8119 


8169 


8219 


8269 


8320 


8370 


8420 


8470 


50 


868 


8520 


8570 


8620 


8670 


8720 


8770 


8820 


8870 


8920 


8970 


5° 


869 


9020 


9070 


9120 


9170 


9220 


9270 


9320 


9369 


9419 


9469 


5° 


870 


93 95 J 9 


93 95 6 9 


939619 


93 9669 


93 9719 


93 97 6 9 


939819 


93 9869 


939918 


93 9968 


5° 


871 


940018 


94 0068 


940118 


940168 


940218 


940267 


940317 


94 0367 


940417 


94 0467 


5° 


872 


0516 


0566 


0616 


0666 


0716 


0765 


0815 


0865 


0915 


0964 


5° 


873 


1014 


1064 


1114 


1163 


1213 


1263 


1313 


1362 


1412 


1462 


5° 


874 


1511 


1561 


1611 


1660 


1710 


1760 


1809 


1859 


1909 


1958 


5° 


875 


94 2008 


94 2058 


942107 


942157 


94 2207 


94 2256 


94 2306 


94 2355 


94 2405 


94 2455 


50 


876 


2504 


2554 


2603 


2653 


2702 


2752 


2801 


2851 


2901 


2950 


5° 


877 


3000 


3049 


3°99 


3148 


3198 


3247 


3297 


3346 


3396 


3445 


49 


878 


3495 


3544 


3593 


3643 


3692 


3742 


379i 


3841 


3890 


3939 


49 


879 


3989 


4038 


4088 


4137 


4186 


4236 


4285 


4335 


4384 


4433 


49 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 



LOGARITHMS OF NUMBERS. 



177 



N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 

49 


880 


94 4483 


94 4532 


944581 


944631 


94 4680 


94 47 2 9 


94 4779 


94 4828 


944877 


94 4927 


881 


4976 


5° 2 5 


5°74 


5 I2 4 


5 r 73 


5222 


5272 


5321 


537° 


5419 


49 


882 


5469 


5518 


5567 


5616 


5665 


5715 


5764 


5813 


5862 


5912 


49 


883 


5961 


6010 


6059 


6108 


6i57 


6207 


6256 


6305 


6354 


6403 


49 


884 


6452 


6501 


6551 


6600 


6649 


6698 


6747 


6796 


6845 


6894 


49 


885 


94 6943 


94 6992 


94 7°4i 


94 7°9o 


947140 


947189 


94 7238 


94 7287 


94 7336 


94 7385 


49 


886 


7434 


7483 


7532 


7581 


7630 


7679 


7728 


7777 


7826 


7875 


49 


887 


7924 


7973 


8022 


8070 


8119 


8 J 68 


8217 


8266 


8315 


8364 


49 


888 


8413 


8462 


8511 


8560 


8609 


8657 


8706 


8755 


8804 


8853 


49 


889 


8902 


8951 


8999 


9048 


9097 


9146 


9195 


9244 


9292 


934i 


49 


890 


94 9390 


94 9439 


94 9488 


94 9536 


94 9585 


94 9634 


94 9683 


94 9731 


94 9780 


949829 


49 


891 


9878 


9926 


9975 


95 °° 2 4 


95 °°73 


95 OI2 i 


95 OI 7° 


950219 


95 0267 


95°3i6 


49 


8g2 


95 0365 


95 °4i4 


95 0462 


05 1 1 


0560 


0608 


0657 


0706 


0754 


0803 


49 


893 


0851 


0900 


0949 


0997 


1046 


io95 


1 143 


1192 


1240 


1289 


49 


894 


1338 


1386 


1435 


H83 


1532 


1580 


1629 


1677 


1726 


1775 


49 


895 


95 lg 23 


95 1872 


95 J 9 2 o 


95 ^69 


95 2017 


95 2066 


95 21 14 


95 2163 


95 221 1 


95 2260 


48 


896 


2308 


235 6 


2405 


2453 


2502 


2550 


2599 


2647 


2696 


2744 


48 


897 


2792 


2841 


2889 


2938 


2986 


3034 


3083 


3131 


3180 


3228 


48 


898 


3276 


3325 


3373 


3421 


3470 


35i8 


3566 


361.5 


3663 


37" 


48 


899 


3760 


3808 


3856 


3905 


3953 


4001 


4049 


4098 


4146 


4194 


48 


900 


95 4243 


95 4291 


95 4339 


95 4387 


95 4435 


95 4484 


95 4532 


95 458o 


95 4628 


95 4677 


48 


go 1 


4725 


4773 


4821 


4869 


4918 


4966 


5 OI 4 


5062 


5110 


5158 


48 


902 


52 oZ 


5255 


5303 


535i 


5399 


5447 


5495 


5543 


5592 


5640 


48 


903 


5688 


5736 


5784 


5832 


5880 


5928 


5976 


6024 


6072 


6120 


48 


904 


6168 


6216 


6265 


6313 


6361 


6409 


6457 


65 5 


6553 


6601 


48 


905 


95 66 49 


95 66 97 


95 6745 


95 6 793 


95 6840 


95 6888 


95 6 936 


95 6984 95 7°32 


95 7080 


48 


906 


7128 


7176 


7224 


7272 


7320 


7368 


7416 


7464 


7512 


7559 


48 


907 


7607 


7655 


7703 


775 1 


7799 


7847 


7894 


7942 


7990 


8038 


48 


go8 


8086 


8i34 


8181 


8229 


8277 


8325 


8373 


8421 


8468 


8516 


48 


909 


8564 


8612 


8659 


8707 


8755 


8803 


8850 


8898 


8946 


8994 


48 


910 


95 9041 


95 9089 


95 9137 


959185 


95 9232 


95 9280 


95 9328 


95 9375 


95 9423 


95 947i 


48 


gn 


95i8 


9566 


9614 


9661 


9709 


9757 


9804 


9852 


9900 


9947 


48 


gi2 


9995 


96 0042 


96 0090 


960138 


960185 


960233 


960280 


960328 


960376 


96 0423 


48 


9i3 


960471 


0518 


0566 


0613 


0661 


0709 


0756 


0804 


0851 


0899 


48 


9M 


0946 


0994 


1041 


1089 


1136 


1 184 


1231 


1279 


1326 


1374 


48 


9i5 


96 1421 


96 1469 


96 1516 


961563 


96 1611 


96 1658 


96 1706 


96i753 


96 1 801 


96 1848 


47 


gi6 


1895 


1943 


1990 


2038 


2085 


2132 


2180 


2227 


2275 


2322 


47 


9i7 


2369 


2417 


2464 


25 1 1 


2559 


2606 


2653 


2701 


2748 


2795 


47 


918 


2843 


2890 


2937 


2985 


3032 


3°79 


3126 


3i74 


3221 


3268 


47 


gig 


33i6 


3363 


34io 


3457 


3504 


3552 


3599 


3646 


3693 


374i 


47 


g2o 


96 3788 


96 3835 


96 3882 


96 3929 


96 3977 


96 4024 


964071 


964118 


964165 


964212 


47 


g2i 


4260 


4307 


4354 


4401 


4448 


4495 


4542 


459o 


4637 


4684 


47 


922 


473i 


4778 


4825 


4872 


4919 


4966 


5 OI 3 


5061 


5108 


5 J 55 


47 


923 


5202 


5249 


5296 


5343 


5390 


5437 


5484 


553i 


5578 


5625 


47 


924 


5672 


5719 


5766 


5813 


5860 


5907 


5954 


6001 


6048 


6095 


47 


925 


96 6142 


966189 


966236 


966283 


96 6329 


96 6376 


96 6423 


96 6470 


966517 


96 6564 


47 


g26 


661 1 


6658 


6705 


6752 


6799 


6845 


6892 


6939 


6986 


7033 


47 


927 


7080 


7127 


7173 


7220 


7267 


73H 


736i 


7408 


7454 


75 QI 


47 


928 


7548 


7595 


7642 


7688 


7735 


7782 


7829 


7875 


7922 


7969 


.47 


929 


8016 


8062 


8109 


8156 


8203 


8249 


8296 


8343 


8390 


8436 


47 


930 


96 8483 


968530 


968576 


96 8623 


96 8670 


968716 


96 8763 


968810 


968856 


96 8903 


47 


93i 


8950 


8996 


9°43 


9090 


9136 


9183 


9229 


9276 


9323 


9369 


47 


932 


9416 


9463 


95°9 


9556 


9602 


9649 


9695 


9742 


9789 


9835 


47 


933 


9882 


9928 


9975 


970021 


97 0068 


97 01 14 


970161 


97 0207 


97 ° 2 54 


97 0300 


47 


934 


97 °347 


97 °393 


970440 


0486 


0533 


0579 


0626 


0672 


0719 


0765 


46 


935 


970812 


970858 


97 0904 


970951 


970997 


97 IQ 44 


97 1090 


97 "37 


971183 


97 1229 


46 


936 


1276 


1322 


1369 


1415 


1461 


1508 


1554 


1 601 


1647 


1693 


46 


937 


1740 


1786 


1832 


1879 


1925 


1971 


2018 


2064 


2110 


2157 


46 


938 


2203 


2249 


2295 


2342 


2388 


2434 


2481 


2527 


2573 


2619 


46 


939 


2666 


2712 


2758 


2804 


2851 


2897 


2943 


2989 


3035 


3082 


46 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 



178 



LOGARITHMS OF NUMBERS. 



N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 


940 


973128 


973174 


97 3220 


97 3 266 


97 3313 


97 3359 


97 3405 


973451 


97 3497 


97 3543 


46 


941 


3590 


3636 


3682 


3728 


3774 


3820 


3866 


3913 


3959 


4005 


46 


942 


4051 


4097 


4H3 


4189 


4235 


4281 


4327 


4374 


4420 


4466 


46 


943 


45 12 


4558 


4604 


4650 


4696 


4742 


4788 


4834 


4880 


4926 


46 


944 


4972 


5018 


5064 


5110 


5 J 5 6 


5202 


5248 


5294 


5340 


5386 


46 


945 


97 5432 


97 5478 


97 5524 


97 557° 


975616 


97 5662 


97 5707 


97 5753 


97 5799 


97 5845 


46 


946 


5891 


5937 


5983 


6029 


6075 


6121 


6167 


6212 


6258 


6304 


46 


947 


635° 


6396 


6442 


6488 


6533 


6579 


6625 


6671 


6717 


6763 


46 


948 


6808 


6854 


6900 


6946 


6992 


7037 


7083 


7129 


7175 


7220 


46 


949 


7266 


7312 


7358 


7403 


' 7449 


7495 


754i 


7586 


7632 


7678 


46 


95o 


97 7724 


97 7769 


97 78i5 


97 7861 


97 7906 


97 7952 


97 7998 


97 8043 


97 8089 


97 8135 


46 


95i 


8181 


8226 


8272 


8317 


8363 


8409 


8454 


8500 


8546 


8591 


46 


952 


8637 


8683 


8728 


8774 


8819 


8865 


891 1 


8956 


9002 


9047 


46 


953 


9093 


9138 


9184 


9230 


9275 


9321 


9366 


9412 


9457 


95°3 


46 


954 


9548 


9594 


9639 


9685 


973° 


9776 


9821 


9867 


9912 


9958 


46 


955 


98 0003 


98 0049 


98 0094 


98 0140 


980185 


980231 


980276 


98 0322 


98 0367 


980412 


45 


956 


0458 


0503 


0549 


0594 


0640 


0685 


0730 


0776 


0821 


0867 


45 


957 


0912 


°957 


1003 


1048 


1093 


"39 


1 184 


1229 


1275 


1320 


45 


958 


1366 


1411 


1456 


1501 


1547 


1592 


1637 


1683 


1728 


1773 


'45 


959 


1819 


1864 


1909 


1954 


2000 


2045 


2090 


2135 


2181 


2226 


45 


960 


98 2271 


982316 


98 2362 


98 2407 


98 2452 


98 2497 


98 2543 


98 2588 


98 2633 


98 2678 


45 


961 


2723 


2769 


2814 


2859 


2904 


2949 


2994 


3040 


3085 


3130 


45 


962 


3175 


3220 


3265 


33io 


335 6 


3401 


3446 


349i 


3536 


358i 


45 


963 


3626 


3671 


37i6 


3762 


3807 


3852 


3897 


3942 


3987 


4032 


45 


964 


4077 


4122 


4167 


4212 


4257 


4302 


4347 


4392 


4437 


4482 


45 


965 


984527 


98 4572 


984617 


98 4662 


98 47°7 


984752 


98 4797 


98 4842 


98 4887 


98 4932 


45 


966 


4977 


5022 


5067 


5112 


5*57 


5202 


5247 


5292 


5337 


5382 


45 


967 


5426 


547i 


55i6 


556i 


5606 


5651 


5696 


574i 


5786 


5830 


45 


968 


5875 


5920 


5965 


6010 


6055 


6100 


6144 


6189 


6234 


6279 


45 


969 


6324 


6369 


6413 


6458 


6503 


6548 


6593 


6637 


6682 


6727 


45 


970 


98 6772 


986817 


98 6861 


98 6906 


98 6951 


98 6996 


98 7040 


98 7085 


98 7130 


987175 


45 


971 


7219 


7264 


7309 


7353 


7398 


7443 


7488 


7532 


7577 


7622 


45 


972 


7666 


771 1 


7756 


7800 


7845 


7890 


7934 


7979 


8024 


8068 


45 


973 


8113 


8157 


8202 


8247 


8291 


8336 


8381 


8425 


8470 


8514 


45 


974 


8559 


8604 


8648 


8693 


8737 


8782 


8826 


8871 


8916 


8960 


45 


975 


98 9005 


98 9049 


98 9094 


989138 


989183 


98 9227 


98 9272 


989316 


98 9361 


98 9405 


45 


976 


945° 


9494 


9539 


9583 


9628 


9672 


9717 


9761 


9806 


9850 


44 


977 


9895 


9939 


9983 


99 0028 


990072 


99 01 1 7 


99 0161 


99 0206 


99 0250 


99 0294 


44 


978 


99 0339 


99 0383 


99 0428 


0472 


0516 


0561 


0605 


0650 


0694 


0738 


44 


979 


0783 


0827 


0871 


0916 


0960 


1004 


1049 


1093 


"37 


1 182 


44 


980 


99 1226 


99 1270 


99I3I5 


99 1359 


99 1403 


99 1448 


99 1492 


99 1536 


99 1580 


99 1625 


44 


981 


1669 


1713 


1758 


1802 


1846 


1890 


1935 


1979 


2023 


2067 


44 


982 


2111 


2156 


2200 


2244 


2288 


2333 


2377 


2421 


2465 


2509 


44 


983 


2554 


2598 


2642 


2686 


2730 


2774 


2819 


2863 


2907 


295 1 


44 


984 


2995 


3039 


3083 


3^7 


3172 


3216 


3260 


33°4 


3348 


3392 


44 


985 


99 3436 


99 348o 


99 3524 


99 3568 


99 3613 


99 3657 


99 37 QI 


99 3745 


99 3789 


99 3833 


44 


986 


3877 


3921 


3965 


4009 


4053 


4097 


4141 


4185 


4229 


4273 


44 


987 


4317 


436i 


4405 


4449 


4493 


4537 


4581 


4625 


4669 


47 x 3 


44 


988 


4757 


4801 


4845 


4889 


4933 


4977 


5021 


5065 


5108 


5i5 2 


44 


989 


5 X 96 


5240 


5284 


5328 


5372 


54i6 


546o 


5504 


5547 


559i 


44 


990 


99 5 6 35 


99 5 6 79 


99 5723 


99 5767 


99 581 1 


99 5854 


99 5898 


99 5942 


99 5986 


99 6030 


44 


991 


6074 


6117 


6161 


6205 


6249 


6293 


6337 


6380 


6424 


6468 


44 


992 


6512 


6555 


6599 


6643 


6687 


6731 


6774 


6818 


. 6862 


6906 


44 


993 


6949 


6993 


7037 


7080 


7124 


7168 


7212 


7255 


7299 


7343 


44 


994 


7386 


7430 


7474 


7517 


756i 


7605 


7648 


7692 


7736 


7779 


44 


995 


99 7823 


99 7867 


99 79io 


99 7954 


99 7998 


99 8041 


99 8085 


99 8129 


998172 


99 8216 


44 


996 


8259 


8303 


8347 


8390 


8434 


8477 


8521 


8564 


8608 


8652 


44 


997 


8695 


8739 


8782 


8826 


8869 


8913 


8956 


9000 


9043 


9087 


44 


998 


9i3i 


9174 


9218 


9261 


9305 


9348 


9392 


9435 


9479 


9522 


44 


999 


9565 


9609 


9652 


9696 


9739 


9783 


9826 


9870 


9913 


9957 


43 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 |D. 



TABLE XVIII 

LOGARITHMIC SINES, COSINES, TANGENTS 
[ AND COTANGENTS, 

FOR EVERY 

DEGREE AND MINUTE FROM 0° TO W\ 



180 LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 

0° 



M. 



Sin, 



D. 1' 



Cos. 



D.l' 



Tan. 



D. 1". 



Cot, 



o 

I 


— 00 
6.463 726 


2 


.764 756 


3 


.940 847 


4 


7.065 786 


5 


7.162 696 


6 


.241 877 


7 


.308 824 


8 


.366816 


9 


.417 968 


10 


7.463 726 


ii 


.505118 


12 


.542 906 


13 


.577 668 


14 


•609 853 


15 


7.639816 


16 


.667 845 


17 


.694173 


18 


.718997 


19 


.742478 


20 


7764 754 


21 


785 943 


22 


T .806 146 


23 


.825451 


24 


•843 934 


25 


7.861 662 


26 


.878 695 


27 


.895 085 


28 


.910879 


29 


.926 119 


30 


7.940 842 


3i 


•955 °82 


32 


.968 870 


33 


.982 233 


34 


•995 : 98 


35 


8.007 787 


3b 


.020021 


37 


.031 919 


3» 


.043 501 


39 


.054 781 


40 


8.065 776 


41 


.076 500 


42 


.086 965 


43 


.097 183 


44 


.107 167 


45 


8.116926 


46 


.126471 


47 


.135810 


48 


•144 953 


49 


•153 907 


50 


8.162 681 


5i 


.171 280 


52 


•179 713 


53 


.187985 


54 


.196 102 


55 


8.204 070 


56 


.211 895 


57 


.219581 


58 


.227 134 


59 


•234 557 


60 


8.241 855 



5017.17 
2934.85 

2082.32 
1615.17 

1319.68 
1115.78 

966.53 
852.53 
762.63 

689.87 

629.80 

579-37 
536-42 
499.38 

467-15 
438.80 

4I3-73 
391-35 
37I-27 

353-15 
336-72 

321.75 
308.05 

295-47 
283.88 

273-I7 
263.23 
254.00 
245.38 

237-33 
229.80 
222.72 
216.08 
209.82 
203.90 
198.30 

193-03 
188.00 
183.25 

178.73 
174.42 
170.30 
166.40 
162.65 
159.08 

I55-65 
152.38 
149.23 
146.23 

I43-32 
140.55 
I37-87 
135-28 
132.80 

130.42 
128.10 
125.88 
123.72 
121.63 



10.000000 
.000 000 
.000 000 
.000 000 
.000 000 
10.000 000 

9.999 999 
.999 999 
•999 999 
•999 999 

9.999 998 
-999 998 
-999 997 
•999 997 
•999 996 

9.999 996 
•999 995 
•999 995 
•999 994 
•999 993 

9-999 993 
•999 992 
.999991 
.999 990 
•999 989 

9.999 989 
•999 988 
•999 987 
.999 986 

•999 985 

9-999 983 
.999 982 

-999 981 
.999 980 

•999 979 
9.999 977 
•999 976 
•999 975 
•999 973 
-999 972 
9.999971 

■999 969 
.999 968 
•999 966 
.999 964 

9.999 963 
•999 961 
•999 959 
•999 958 
•999 956 

9-999 954 

.999952 

-999 95° 
•999 948 
•999 946 

9.999 944 
.999 942 
.999 940 
.999 938 
•999 936 

9-999 934 



.00 
.00 
.00 
.00 
.02 
.00 
.00 
.00 
.02 
.00 
.02 
.00 
.02 
.00 
.02 
.00 
.02 
.02 
.00 
.02 
.02 
.02 
.02 
.00 
.02 
.02 
.02 
.02 
•03 
.02 
.02 
.02 
.02 
•03* 
.02 
.02 

•03 
.02 
.02 

•03 
.02 

•03 
•03 
.02 

•03 
•03 
.02 

.03 
•03 
•03 
•03 
•03 
•03 
•03 
•03 
•03 
.03 
•03 
•03 



6.463 726 


•764 756 


■940 847 


7.065 786 


7.162 696 


.241 878 


.308 825 


.366817 


.417970 


7.463 727 


.505 120 


.542 909 


.577 672 


.609 857 


7.639 820 


.667 849 


.694179 


.719003 


.742 484 


7.764 761 


•785 95 1 


.806 155 


.825 460 


.843 944 


7.861 674 


.878 708 


•895 099 


.910 894 


.926 134 


7.940 858 


•955 IO ° 


.968 889 


.982 253 


•995 2I 9 


8.007 809 


.020 044 


•031 945 


•043 5 2 7 


.054 809 


8.065 806 


.076 531 


.086 997 


.097 217 


.107203 


8. 1 16 963 


.126 510 


.135851 


.144996 


•153 952 


8.162 727 


.171 328 


.179763 


.188036 


.196 156 


8.204 126 


•211 953 


.219 641 


•227 195 


.234 621 


8.241 921 



5017.17 
2934-85 

2082.32 

1615.17 

1319.70 

1115.78 

966.53 

852.55 

762.62 

689.88 

629.82 

579-38 
536.42 
499.38 

467-15 
438.83 
413-73 

391-35 
371.28 

353-17 
336-73 
321.75 
308.07 

295-5° 
283.90 
273.18 
263.25 
254.00 
245.40 

237-37 
229.82 
222.73 
216.10 
209.83 
203.92 

198.35 
19303 
188.03 
183.28 
178.75 
174-43 
I70-33 
166.43 
162.67 
159.12 
155.68 
152.42 
149.27 
146.25 

143.35 
140.58 
137.88 
135-33 
132.83 

130.45 
128.13 
125.90 

12377 
121.67 



— 00 


60 


3.536 274 


59 


.235 244 


58 


•059 153 


57 


2.934 214 


56 


2.837 304 


55 


.758 122 


54 


.691 175 


53 


•633 183 


52 


.582 030 


5i 


2.536 273 


50 


.494 880 


49 


.457091 


48 


.422 328 


47 


•390 143 


46 


2.360 180 


45 


.332 151 


44 


•305 821 


43 


.280 997 


42 


.257 5 J 6 


41 


2.235 239 


40 


.214049 


39 


.193845 


38 


.174540 


37 


.156056 


36 


2.138326 


35 


.121 292 


34 


.104901 


33 


.089 106 


32 


.073 866 


3i 


2.059 142 


30 


.044 900 


29 


.031 in 


28 


.017747 


27 


.004 781 


26 


1.992 191 


25 


•979 956 


24 


.968055 


23 


.956 473 


22 


■945 J 9i 


21 


1-934 194 


20 


.923 469 


19 


.913003 


18 


.902 783 


17 


.892 797 


16 


1.883037 


15 


•873 490 


14 


.864 149 


13 


.855 004 


12 


.846 048 


11 


1-837273 


10 


.828 672 


9 


.820 237 


8 


.811 964 


7 


.803 844 


6 


1.795 874 


5 


.788 047 


4 


.780 359 


3 


.772 805 


2 


765 379 


1 


1.758079 






Cos. 



D. 1". 



Sin. 



D. 1". 
"89° 



Cot. 



D. 1' 



Tan. 



LOGARITHMIC SIXES, COSINES, TANGENTS, AND COTANGENTS. 181 

1° 



Sin. 



o 


8.241 855 


I 


•249 033 


2 


.256 094 


3 


.263 042 


4 


.269 881 


5 


8.276614 


6 


.283 243 


7 


.289 773 


8 


.296 207 


9 


.302 546 


ito 


8.308 794 


ii 


.314 954 


12 


.321 027 


13 


.327016 


14 


•332 924 


15 


8.338 753 


16 


•344 5°4 


17 


.350181 


18 


•355 783 


19 


•361 315 


20 


8.366 777 


21 


.372171 


22 


•377 499 


23 


.382 762 


24 


•387 962 


25 


8.393 101 


26 


•398 179 


27 


.403 199 


28 


.408 161 


29 


.413068 


30 


8.417 919 


31 


.422717 


32 


.427 462 


33 


.432 156 


34 


.436 800 


35 


8.441 394 


36 


•445 94i 


37 


.450 440 


38 


•454 893 


39 


•459 3 QI 


40 


8.463 665 


4i 


.467 985 


42 


.472 263 


43 


.476 498 


44 


.480 693 


45 


8.484 848 


46 


.488 963 


47 


•493 040 


48 


•497 °7 8 


49 


.501 080 


50 


8-505 °45 


5i 


.508 974 


52 


.512867 


53 


.516 726 


54 


•520551 


55 


8.524 343 


56 


.528 102 


57 


.531 828 


5» 


•535 5 2 3 


59 


•539 186 


60 


8.542819 



D. 1' 



Cos. 



119.63 

117.68 

115.80 

113.98 

112.22 

110.48 

108.83 

107.23 

105.65 

104.13 

102.67 

101.22 

99.82 

98.47 

97-15 

95.85 

94.62 

93-37 
92.20 
91.03 

89.90 
88.80 
87.72 
86.67 
85.65 
84.63 
83.67 
82.70 
81.78 
80.85 

79-97 
79.08 

78.23 
77.40 

76.57 
75-78 
74.98 
74.22 
73-47 
72-73 
72.00 
7!-3o 
70.58 
69.92 
69.25 
68.58 

67-95 
67.30 
66.70 
66.08 

6548 
64.88 
64.32 

63.75 
63.20 

62.65 
62.10 
61.58 
61.05 
60,55 

D. 1", 



Cos. 



9-999 934 
•999 932 
•999 929 
•999 927 
•999 925 

9.999 922 
•999 920 
.999918 

.999 915 
•999 9*3 

9.999910 
•999 907 
•999 905 
•999 902 
•999 899 

9.999 897 
.999 894 
.999 891 
.999 888 
•999 885 

9.999 882 
•999 879 
•999 876 
•999 873 
•999 870 

9.999 867 
.999864 
.999 861 
•999 858 
•999 854 

9.9.99851 
.999 848 
•999 844 
•999 841 
•999 838 

9.999 834 
•999 831 
•999 827 
.999 824 
.999 820 

9.999 816 
•999 8i3 
•999 809 
•999 805 
.999 801 

9.999 797 
•999 794 
•999 790 
•999 786 
•999 782 

9.999 778 
•999 774 
•999 769 
•999 765 
•999 761 

9-999 757 
•999 753 
.999 748 

•999 744 

•999 740 

9-999 735 

Sin. 



D. 1". 



.03 
•°5 
.03 
•03 
•05 
•03 
•03 
•°5 
•03 
•°5 
•05 
•03 
•05 
•05 
•03 

•05 
•05 
•05 
•05 
.05 
•05 
•°5 
•05 
•05 
•°5 
•°5 
•05 
•°5 
.07 

•°5 

•05 
.07 

•°5 
•05 
.07 

•°5 
.07 

•05 
.07 

•07 

•05 
.07 

•°7 
.07 
.07 

.05 
.07 
.07 
.07 
.07 
.07 
.08 
.07 
.07 
.07 

•07 
.08 
.07 
.07 
.08 



D. 1". 

88° 



Tan, 



5.241 921 
.249 102 
.256 165 
.263 115 
.269 956 

8.276 691 
.283 323 
.289 856 
.296 292 
.302 634 

8.308 884 
.315046 
.321 122 
.327114 
•333 025 

8.338856 
.344 610 
•35° 289 
•355 895 
.361 430 

8.366 895 
.372 292 
.377622 
.382 889 
.388092 

8-393 234 
.398315 
•403 338 
.408 304 

•413 213 

8.418068 
.422 869 
.427 618 

•432315 
•436 962 
8.441 560 
.446 no 
.450613 

•455 070 
.459481 

8.463 849 
.468 1 72 

•472 454 
.476 693 
.480 892 
8.485 050 
.489 170 
•493 250 
•497 293 
.501 298 

8.505 267 
.509 200 
.513098 
.516961 
.520 790 

8.524586 
.528 349 
.532080 

•535 779 

•539 447 

8-543084 

Cot. 



D. 1". 



119.68 
117.72 
115.83 
114.02 
112.25 
110.53 
108.88 
107.27 
105.70 
104.17 
102.70 
101.27 
99.87 
98.52 
97.18 

95-90 
94-65 
93-43 
92.25 
91.08 

89-95 
88.83 
87.78 
86.72 
85.70 
84.68 
83-72 
82.77 
81.82 
80.92 
80.02 

79-15 

78.28 

77-45 
76.63 

75-83 
75-o5 
74.28 

73-52 
72.80 

72.05 
71-37 
70.65 
69.98 
69.30 
68.67 
68.00 
67-38 
66.75 
66.15 

65-55 
64.97 

64.38 
63.82 

63.27 
62.72 
62.18 
61.65 
61.13 
60.62 

D. 1". 



Cot. 



1.758079 
.750 898 

743 835 
736885 
730 044 

723 309 
716677 
710 144 
703 708 
697 366 
691 116 
684 954 
678 878 
672886 
666975 
661 144 

655 390 
649 711 
644 105 
638570 
633 105 
627 708 
622 378 
617 in 
611 908 
606 766 
601 685 
596 662 
591 696 
586 787 

581 932 

577 131 
572382 
567 685 
563038 

558 440 
553890 
549 387 
544 93o 
540 5 J 9 

536 151 
531 828 

527 546 
523 307 
519 108 

5^95° 
510830 
506 750 
502 707 
498 702 

494 733 

490 800 
4S6 902 

483 039 
479 210 

475 414 
47 1 651 
467 920 
464 221 
460 553 
456916 

Tan. 



60 
59 
58 
57 
56 

55 
54 
53 
52 
5i 
50 
49 
48 

47 
46 

45 
44 
43 
42 
4i 
40 
39 
38 
37 
36 

35 
34 
33 
32 
3i 
30 
29 
28 

27 
26 

25 
24 
23 
22 
21 
20 

19 
18 

17 
16 

15 
14 
13 
12 
11 



9 

8 

7 
6 

5 

4 
3 

2 
1 
o 



182 LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 

2° 



M. 



o 

i 

2 

3 
4 

5 
6 

7 
8 

9 

10 

n 

12 

13 

15 
16 

17 

18 

19 

20 
21 
22 
23 

24 

25 
26 

27 

28 
29 

30 
31 
32 

33 
34 

35 
36 
37 
38 
39 
40 
4i 
42 
43 
44 

45 
46 

47 
48 

49 
50 
5i 
52 
53 
54 

55 
56 
57 
58 
59 
60 



Sin. 



3.542 819 
.546 422 
•549 995 
•553 539 
•557 °54 
5.560 540 
•5 6 3 999 
•567 431 
.570836 
.574214 

^•577 5 6 6 
.580 892 
.584 193 
.587 469 
.590 721 

^•593 948 
•597 J 5 2 
.600 332 
.603 489 
.606 623 
5.609 734 
.612 823 
.615 891 
.618937 
.621 962 
5.624 965 
.627 948 
.630 911 

•633854 
.636776 

5.639 680 
.642 563 
.645 428 
.648 274 
.651 102 
5.653 911 
.656 702 

•659 475 
.662 230 
.664 968 
5.667 689 
•670 393 
.673 080 

•675 751 
.678 405 

5.68i 043 
.683 665 
.686272 
.688 863 
.691 438 

3.693 998 
•696 543 
.699073 
.701 589 
.704090 

3.706577 
.709049 
.711507 
•7 I 3 95 2 
.716383 

3.718800 

Cos. 



D. 1' 



60.05 

59-55 
59-07 
58.58 
58.10 

57-65 
57.20 

56.75 
56-30 
55-87 
55-43 
55.02 
54.60 
54.20 
53.78 

53-40 
53.00 
52.62 

5 2 . 2 3 
51.85 

51.48 

5o-77 
50.42 

5°-°5 
49.72 
49-38 
49.05 
48.70 
48.40 
48.05 
47-75 
47-43 
47-13 
46.82 

46.52 
46.22 
45-92 
45.63 
45-35 

45-°7 
44.78 

44-52 
44-23 
43-97 
43-7o 
43-45 
43-i8 
42.92 
42.67 
42.42 
42.17 

41-93 
41.68 

41-45 
41.20 
40.97 

40.75 
40.52 
40.28 

D. 1". 



Cos. 



9-999 735 
•999 73i 
.999 726 
.999 722 
.999717 

9-999 7I3 
•999 7°8 
.999 704 

•999 699 
•999 694 

9.999 689 
•999 685 
.999 680 
•999 675 
•999 670 

9.999 665 
.999 660 
•999 655 
•999 650 
•999 645 

9.999 640 
•999 635 
•999 629 
.999 624 
.999619 

9.999 614 
.999 608 
•999 603 
•999 597 
•999 592 

9.999 586 
•999 581 
•999 575 
•999 57o 
•999 564 

9-999 558 
•999 553 
•999 547 
-999 54i 
•999 535 

9.999 529 

•999 5 2 4 
.999518 
.999512 
•999 5°6 

9.999 500 
•999 493 
•999 487 
.999 481 
•999 475 

9.999 469 

•999 463 
.999 456 

•999 45° 
•999 443 
9-999 437 
.999 431 
.999 424 
.999418 
.999411 

9999 404 
Sin. 



D. 1' 



D. 1". 

87° 



Tan. 



8-543 084 
.546 691 
.550268 
•553 817 
•557 336 

8.560 828 
.564 291 
•567 727 
•571 137 
.574 520 

8.577 877 
.581 208 

•584 5 J 4 
•587 795 
•59i 051 

8.594 283 
•597 492 
.600 677 
•603 839 
.606 978 

8.610094 

.613 189 
.616 262 
.619313 
•622 343 

8.625 352 
.628 340 
.631 308 
•634 256 
.637 184 

8.640 093 
.642 982 

.645 853 
.648 704 

.651 537 
8.654352 
.657 149 
.659 928 
.662 689 
•665 433 
8.668 160 
.670 870 

•673 563 
•676 239 
.678 900 

8.681 544 
.684 172 
.686 784 
.689 381 
•691 963 

8.694 529 
.697081 
.699 617 

.702 139 
.704 646 

8.707 140 
.709 618 
.712083 

.7H534 
.716972 

8.7i9 396 

Cot. 



D. 1' 



60.12 
59.62 

59-15 
58.65 
58.20 

57-72 
57.27 
56.83 
56.38 

55-95 

55-52 

55-1° 
54.68 

54.27 
53-87 
5348 
53-o8 
52.70 
52-32 
51-93 
51.58 
51.22 
50.85 
50.50 
50.15 
49.80 
49-47 
49-13 
48.80 
48.48 

48.15 
47-85 
47-52 
47.22 
46.92 
46.62 
46.32 
46.02 

45-73 
4545 

45- r 7 
44.88 
44.60 

44-35 
44.07 

43.80 

43-53 
43.28 

43-03 
42.77 

42.53 
42.27 
42.03 
41.78 
41-57 
41.30 
41.08 
40.85 
40.63 
40.40 

D. 1". 



Cot. 



456916 
453 309 
449 732 
446 183 
442 664 



439 
435 
432 
428 

425 
422 
418 

415 
412 
408 

405 
402 

399 
396 
393 

389 
386 

383 
380 

377 
374 
37i 
368 

365 
362 

359 
357 
354 
35i 
348 

345 
342 
34o 
337 
334 
33i 
329 
326 

323 
321 

3i8 
3i5 
3*3 
310 

308 

305 
302 

3°o 
297 

295 
292 
290 

287 

285 
283 

280 



172 

7°9 
273 
863 
480 

123 

792 
486 
205 
949 
717 
508 

323 
161 
022 
906 
811 
738 
687 

657 
648 
660 
692 

744 
816 

907 
018 
147 
296 
463 
648 
851 
072 
3ii 
567 
840 
130 

437 
761 
100 

456 
828 
216 
619 
037 

471 
919 

383 
861 

354 

860 
382 
917 
466 
028 
604 



Tan. 



LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 183 

3° 



o 


8.718 800 


I 


.721 204 


2 


•7 2 3 595 


3 


.725 972 


4 


•7 2 8 337 


5 


8.730 688 


6 


•733027 


7 


•735 354 


8 


•737 667 


9 


•739 969 


IO 


8.742 259 


ii 


•744 536 


12 


.746 802 


13 


•749 055 


14 


.751 297 


15 


8.753 528 


16 


•755 747 


17 


•757 955 


18 


.760 151 


19 


•762337 


20 


8.764 511 


21 


.766 675 


22 


.768 828 


23 


.770970 


24 


.773101 


25 


8.775 223 


26 


•777 333 


27 


•779 434 


28 


.781 524 


29 


.783605 


30 


8.785 675 


31 


.787 736 


32 


.789 787 


33 


.791 828 


34 


•793 859 


35 


8.795 881 


36 


•797 894 


37 


•799 897 


38 


.801 892 


39 


.803 876 


40 


8.805 852 


4i 


.807819 


42 


•809 777 


43 


.811 726 


44 


.813667 


45 


8.815 599 


46 


.817 522 


47 


.819436 


48 


.821 343 


49 


.823 240 


50 


8.825 130 


5i 


.827011 


52 


.828 884 


53 


.830 749 


54 


.832 607 


55 


8.834456 


56 


•836 297 


57 


.838 130 


58 


•839 956 


59 


.841 774 


60 


8.843 585 



Cos. 



D. 1' 



40.07 

39-85 
39.62 

39-42 
39-i8 
38.98 
38.78 
38-55 
38.37 
38.17 
37-95 
37-77 
37-55 
37-37 
37-i8 

36.98 
36.80 
36.60 
3643 
36.23 

36.07 
35.88 
35-70 
35-52 
35-37 

35-17 
35.02 

34-83 
34-68 

34-5° 

34-35 
34.18 
34.02 
33-85 
33-7° 

33-55 
33-38 
33-25 
33-07 
32.93 
32.78 
32.63 
32.48 

32.35 
32.20 

32.05 
31.90 

3I-78 
31.62 

31-5° 

31-35 
31.22 
31.08 

30.97 
30.82 

30.68 
30.55 
30-43 
30.30 
30.18 

D. 1". 



Cos. 



9.999 404 
•999 398 
•999 391 
•999 384 
■999 378 

9-999 37 1 
•999 364 
•999 357 
•999 35° 
•999 343 

9-999 336 
•999 329 
•999 322 
•999 315 
•999 308 

9.999 301 
•999 294 
•999 287 
•999 279 
•999 272 

9.999 265 
•999 257 
•999 250 
•999 242 
•999 235 

9.999 227 
.999 220 
.999 212 
•999 205 
•999 197 

9.999 189 
.999 181 

•999 174 
.999 166 

•999 158 
9.999 150 
•999 142 
•999 134 
.999 126 
.999 118 
9.999 no 
.999 102 

•999 094 
.999 086 
.999077 

9.999 069 
.999 061 
•999 053 
•999 044 
•999 036 

9.999 027 

•999 019 
.999010 
.999 002 
•998 993 

9.998 984 
•998 976 
.998 967 
.998958 
•998 95° 

9-998 941 
Sin. 



D. 1' 



Tan. 



D. 1", 

86° 



8.719396 
.721 806 
.724 204 
.726 588 
•728 959 

8-731317 
•733663 

.735 996 
•738317 
.740 626 

8.742 922 
•745 207 
•747 479 
•749 740 
.751 989 

8.754 227 
•756453 

.758 668 
.760 872 
.763 065 
8.765 246 
.767417 

•769 578 
.771 727 
.773 866 

8-775 995 
•778 114 
.780 222 
.782 320 
.784408 

8.786486 
.788554 
.790 613 
.792 662 
•794 7 GI 

8.796731 
•798 752 
.800 763 
.802 765 
.804 758 

8.806 742 
.808717 
.810683 
.812 641 
•814589 

8.816529 
.818461 
.820 384 
.822 298 
.824 205 

8.826 103 
.827 992 
.829 874 
.831 748 
•833613 

8.835471 
•837 32i 
•839 163 
.840 998 
.842 825 

8.844 644 

Cot. 



40.17 
39-97 
39-73 
39-52 
39-3o 
39.10 
38.88 
38.68 
38.48 
38.27 
38.08 

37-87 
37.68 
3748 
37-3o 
37-io 
36.92 
36-73 
36-55 
36-35 
36.18 
36.02 
35-82 
35-65 
35-48 

35-32 
35-J3 
34-97 
34.80 

34.63 

3447 
34-32 
34-15 
33-98 
33-83 
33-68 
33-5 2 
33-37 
33-22 
33-07 
32.92 
32-77 
3 2 -63 
3247 
32.33 
32.20 

32.05 
31.90 
3I-78 
31-63 
31.48 
3*-37 
31-23 
31.08 

30.97 
30.83 
30.70 
30.58 
3045 
30-32 



D. 1". 



Cot. 



280 604 
278 194 
275 796 
273412 
271 041 
268 683 
266337 
264 004 
261 683 
259 374 
257078 
254 793 
252521 
250 260 
248 01 1 

245 773 
243 547 
241 332 
239 128 
236935 
234 754 
232 583 
230 422 
228 273 
226 134 
224 005 
221 886 
219778 
217 680 
215 592 

213 5H 

211 446 

209 387 
207 338 
205 299 
203 269 
201 248 
199237 

197235 
195 242 

193 258, 
191 283 

189 317 
187 359 
185 411 

183 47 1 
181 539 
179 616 
177702 
175 795 

173897 
172 008 
170 126 
168 252 
166387 
164 529 
162 679 
160837 
159002 
157 175 
155 356 
Tan. 



60 
59 
58 
57 
56 
55 
54 
53 
52 
5i 
50 
49 
48 
47 
46 

45 
44 
43 
42 

4i 
40 
39 
38 
37 
36 

35 
34 
33 
32 
3i 
30 
29 
28 

27 
26 

25 

24 

23 
22 
21 



18 

17 
16 

15 
14 
13 
12 
n 



184 LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 

40 



o 
1 
2 
3 
4 

5 
6 

7 
8 

9 
10 
11 
12 
13 
14 

15 
16 

17 
18 

19 
20 
21 
22 

23 

24 

25 

25 

27 
28 
29 

30 
31 
32 
33 
34 

35 
36 
37 
38 
39 
40 

4i 
42 

43 
44 

45 
46 

47 
48 

49 
50 
5i 
52 

53 
54 
55 
56 
57 
58 
59 
60 



Sin. 



5.843 585 
.845 387 
.847 i&3 
.848 971 

•850 751 
L852 525 
•854 291 
.856 049 
.857 801 
.859 546 

:.86i 283 
.863014 

.864 738 
.866455 

.868 165 
L869 868 
.871 565 
•873255 
.874938 
.876615 

5.878 285 
.879 949 
.881 607 
.883 258 
.884 903 
5.886 542 
.888 174 
.889 801 
.891 421 
•893 035 
5.894 643 
.896 246 
.897 842 
.899 432 
.901 017 
5.902 596 
.904 169 

•905 736 
.907 297 
.908 853 
5.910404 
.911 949 
.913 488 
.915 022 
.916550 
5.918073 
.919591 
.921 103 
.922 610 
.924 112 
5.925 609 
.927 100 
.928 587 
.930 068 
•93i 544 

^•933oi5 

•934 48i 
•935 942 
•937 398 
.938850. 

5.940 296 
Cos. 



D. 1". 



30-03 
29-93 
29.80 
29.67 
29-57 

29-43 
29.30 
29.20 
29.08 
28.95 
28.85 

28.73 
28.62 
28.50 
28.38 
28.28 
28.17 
28.05 
27-95 
27-83 

27-73 
27.63 
27.52 
27.42 
27.32 
27.20 
27.12 
27.00 
26.90 
26.80 
26.72 
26.60 
26.50 
26.42 
26.32 
26.22 
26.12 
26.02 

25-93 
25.85 

25-75 
25-65 
25-57 
25-47 
25.38 

25.30 
25.20 
25.12 
25.03 
24-95 
24.85 
24.78 
24.68 
24.60 
24.52 

24-43 
24-35 
24.27 
24.20 
24.10 

D. 1". 



Cos. 



9.998 941 
.998 932 
•998 923 
.998914 
•998 905 

9.998 896 
.998 887 
.998878 
.998 869 
.998 860 

9.998851 
.998 841 
.998832 
•998 823 
.998813 

9.998 804 
.998 795 
•998 785 
•998 776 
.998 766 

9-998 757 
.998 747 
•998 738 
.998 728 
.998 718 

9.998 708 
.998 699 
.998 689 
.998 679 
.998 669 

9.998659 
.998 649 
•998 639 
.998 629 
.998 619 

9.998 609 
•998 599 
.998 589 
•998 578 
.998 568 

9-998 558 
•998 548 

.998 537 

.998527 

.998 516 

9.998 506 

•998 495 
•998 485 
.998 474 
•998 464 

9-998 453 
.998 442 
.998431 
.998 421 
.998410 

9.998 399 
.998 388 
.998 377 
.998 366 
•998 355 

9-998 344 

Sin. 



D. 1' 



D, 1". 

85° 



Tan. 



8.844 644 

•846455 
.848 260 
•850057 
.851 846 
8.853 628 

.855 403 
.857171 

.858932 
.860 686 

8.862433 

•864173 
.865 906 
.867 632 
.869351 

8.871 064 
.872 770 
.874469 
.876 162 
.877 849 

8.879 529 
.881 202 
.882 869 
.884 53o 
.886 185 

8.887 833 
.8S9476 
.891 112 
•892,742 
.894 366 

8.895 984 
.897 596 
•899 203 
.900 803 
•902 398 

8.903 987 
.905570 
•907 147 
.908 719 
.910285 

8.9 1 1 846 
.913401 
.914951 
.916495 
.918034 

8.919568 
.921 096 
.922619 

.924 136 
.925 649 

8.927 156 
.928 658 

•930 155 
.931 647 

•933 134 
8.934616 
.936093 
•937 565 
•939 032 
.940 494 

8-941 952 
Cot. 



D. 1". 



30.18 
30.08 

29-95 
29.82 
29.70 
29.58 
29.47 

29-35 
29.23 
29.12 
29.00 
28.88 

28.77 
28.65 
28.55 

28.43 
28.32 
28.22 
28.12 
28.00 
27.88 
27.78 
27.68 
27.58 
27.47 

27-38 
27.27 
27.17 
27.07 
26.97 
26.87 
26.78 
26.67 
26.58 
26.48 
26.^8 
26.28 
26.20 
26.10 
26.02 
25.92 
25.83 
25-73 
25-65 
25-57 

2547 
25-38 
25.28 
25.22 
25.12 

25-03 

24-95 
24.87 
24.78 
24.70 
24.62 
24-53 
24-45 
24-37 
24.30 

D. 1". 



Cot. 



J55 35 6 

I 53 545 
151 740 

149 943 
148 154 

146372 

144 597 
142 829 
141 068 
139 314 

137 567 
135827 
134094 
132 368 
130649 
128936 
127230 

125 53i 
123838 
122 151 
120 471 
118 798 
117 131 
115 470 
3815 
112 167 
no 524 
108 888 
107 258 
105 634 
104 016 
102 404 
100 797 

099 197 
097 602 

096013 

094 43o 
092 853 
091 281 
?7 l 5 
.088 154 
.0S6 599 
.085 049 
•0S3 505 
~°i 966 

1.080432 
.078 

.077 

•075 
.074 

1.072 
.071 
.069 
.068 
.066 



5904 
1 

4 
■35 1 
1844 
342 
845 
353 
866 

065 384 

063 907 
062 435 
060 968 
°59 5°6 
058 048 

Tan. 



LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 185 

50 - 



M, 



Sin, 






8.940 296 


I 


.941 738 


2 


•943 174 


3 


.944 606 


4 


.946 034 


5 


8.947 456 


6 


.948 874 


7 


.950 287 


8 


.951 696 


9 


•953 100 


10 


8-954 499 


11 


•955 8 94 


12 


•957 2 84 


13 


.958 670 


14 


.960052 


15 


8.961 429 


16 


.962 801 


17 


.964170 


18 


•965 534 


19 


.966 893 


20 


8.968 249 


21 


.969 600 


22 


.970 947 


23 


.972 289 


24 


.973 628 


25 


8.974 962 


26 


.976 293 


27 


.977619 


28 


.978 941 


29 


•980 259 


30 


8.981 573 


3i 


.982 883 


32 


.984 189 


33 


•985 49i 


34 


.986 789 


35 


8.988 083 


36 


•989 374 


37 


.990 660 


38 


•99i 943 


39 


•993 222 


40 


8.994 497 


4i 


.995 768 


42 


.997 036 


43 


.998 299 


44 


•999 5 6 ° 


45 


9.000 816 


46 


.002 069 


47 


.003 318 


48 


.004 563 


49 


.005 805 


50 


9.007 044 


5i 


.008 278 


52 


.009 510 


53 


.010 737 


54 


.011 962 


55 


9.013 182 


56 


.014400 


57 


.015613 


58 


.016 824 


59 


.018031 


60 


9.019 235 




Cos. 



D. 1' 



24.03 
23-93 
23.87 
23.80 
23.70 

23-63 
23-55 
23-48 
23.40 
23-32 
23.25 

23-17 
23.10 

23.03 
22.95 

22.87 
22.82 

22.73 
22.65 
22.60 
22.52 
22.45 
22.37 
22.32 
22.23 
22.18 
22.10 
22.03 
21.97 
21.90 
21.83 
21.77 
21.70 
21.63 
21-57 
21.52 
21.43 
21.38 
21.32 
21.25 
21.18 
21.13 
21.05 
21.02 
20.93 
20.88 
20.82 
20.75 
20.70 
20.65 
20.57 
20.53 
20.45 
20.42 
20.33 
20.30 
20.22 
20.18 
20.12 
20.07 

D. 1". 



Cos. 



9.998 344 
•998 333 
.998 322 
.998311 
.998 300 

9.998 289 
•998 277 
.998 266 
•998 255 
.998 243 

9.998 232 
.998 220 
.998 209 

.998 197 
.998 186 

9.998174 
•998 163 
.998 151 
.998 139 
.998 128 

9.998 116 
.998 104 
.998 092 
.998 080 
.998 068 

9.998 056 
•998 044 
.998 032 
.998 020 
.998 008 

9.997 996 

•997 984 
.997972 

•997 959 
•997 947 

9-997 935 
•997 922 
•997 9io 
•997 897 
•997 885 

9.997 872 
.997 860 
•997 847 
•997 835 
.997 822 

9.997 809 
•997 797 
•997 784 
•997 77 1 
•997 758 

9-997 745 
•997 732 
.997719 

•997 7° 6 
•997 6 93 
9.997 680 
•997 667 
•997 654 
•997 6 4i 
.997 628 

9.997614 



D.l' 



.18 
.18 
.18 
.18 
.18 
.20 
.18 
.18 
.20 
.18 
.20 
.18 
.20 
.18 
.20 
.18 
.20 
.20 
.18 
.20 
.20 
.20 
.20 
.20 
.20 
.20 
.20 
.20 
.20 
.20 
.20 
.20 
.22 
.20 
.20 
.22 
.20 
.22 
.20 
.22 
.20 
.22 
.20 
.22 
.22 
.20 
.22 
.22 
.22 
.22 
.22 
.22 
.22 
.22 
.22 
.22 
.22 
.22 
.22 
-23 



D. 1". 

84° 



8.941 952 
•943 404 
•944 852 
.946 295 
•947 734 

8.949 168 

•95° 597 
.952 021 

•953441 
•954856 

8.956 267 

•957 674 
.959075 
.960 473 
.961 866 

8.963 255 
.964 639 
.966019 

•967 394 
.968 766 

8.970 133 

.971 496 
.972 855 
•974 209 
•975 5 6 ° 
8.976 906 
.978 248 
•979 586 
.980 921 
.982 251 

8.983 577 
.984 899 
.986 217 

•987 532 
.988 842 

8.990 149 
.991451 
.992 750 
•994 045 
•995 337 

8.996 624 
•997 908 
.999 188 

9.000 465 
.001 738 

9.003 007 
.004 272 
.005 534 
.006 792 
.008 047 

9.009 298 
.010 546 
.011 790 
.013031 
.014 268 

9.015 502 
.016 732 

• .017959 
.019 183 
.020 403 

9.021 620 
Cot. 



D, 1" 



24.20 
24.13 
24.05 
23.98 
23.90 
23.82 

23-73 
23.67 
23.58 
23.52 

23-45 
23.35 
23-30 
23.22 

23-15 

23.07 
23.00 
22.92 
22.87 
22.78 
22.72 
22.65 
22.57 
22.52 
22.43 
22.37 
22.30 
22.25 
22.17 
22.10 
22.03 
21.97 
21.92 
21.83 
21.78 
21.70 
21.65 
21.58 
21-53 
21.45 
21.40 

21.33 
21.28 
21.22 
21.15 

21.08 
21.03 
20.97 
20.92 
20.85 
20.80 
20.73 
20.68 
20.62 
20.57 
20.50 

20.45 
20.40 
20.33 
20.28 

D. 1". 



Cot. 



1.058048 


60 


.056 596 


59 


•055 J 48 


58 


•053 705 


57 


.052 266 


56 


1.050832 


55 


•049 403 


54 


•047 979 


53 


.046 559 


52 


.045 r 44 


5i 


1 -043 733 


50 


.042 326 


49 


.040 925 


48 


.039 527 


47 


.038 134 


46 


1-036 745 


45 


.035 361 


44 


.033981 


43 


.032 606 


42 


•031 234 


4i 


1.029 867 


40 


.028 504 


39 


.027 145 


38 


.025 791 


37 


.024 440 


36 


1.023094 


35 


.021 752 


34 


.020 414 


33 


.019079 


32 


.oi7 749 


3i 


1. 01 6 423 


30 


.015 101 


29 


.013 783 


28 


.012468 


27 


.011 158 


26 


1.009 851 


25 


.008 549 


24 


.007 250 


23 


•005 955 


22 


.004 663 


21 


1.003376 


20 


.002 092 


19 


.000 812 


18 


o-999 535 


17 


.998 262 


16 


0.996 993 


IS 


•995 728 


14 


•994 466 


13 


•993 208 


12 


•99i 953 


11 


0.990 702 


10 


.989 454 


9 


.988210 


8 


.986 969 


7 


.985 732 


6 


0.984 498 


5 


.983 268 


4 


.982 041 


3 


.980817 


2 


•979 597 


1 


0.978 380 






Tan. 



186 LOGAEITHMIC SINES, COSINES, TANGENTS, 

6° 



AND COTANGENTS. 



Sin. 



D. 1". 



Cos. 



D. 1", 



Tan. 



D. 1". 



Cot. 



o 
i 

2 

3 
4 

5 
6 

7 
8 

9 

10 

ii 

12 

13 
14 

15 
16 

17 
18 

19 

20 
21 
22 
23 

24 

25 
26 

27 
28 

29 

30 
3i 

32 

33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 

45 
46 

47 
48 

49 
5o 
5i 
52 
53 
54 
55 
56 
57 
58 
59 
60 



).oi9 235 
.020 435 
.021 632 
.022 825 
.024016 
1.025 203 

.026 386 

.027 567 
.028 744 
.029 918 
1.031 089 
.032 257 
.033 421 
.034 582 
•035 74i 
1.036 896 
.038 048 
.039 197 
.040 342 
.041 485 
1.042 625 
.043 762 
.044 895 
.046 026 
.047 154 
1.048 279 
.049 400 
.050519 
•051 635 
.052 749 

'.053 859 
.054 966 
.056071 
.057172 
.058 271 
1.059 367 
.060 460 
.061 551 
.062 639 
.063 724 
1.064 806 
.065 885 
.066 962 
.068 036 
.069 107 
1.070 176 
.071 242 
.072 306 
.073 366 
.074 424 
1.075 4 8 ° 
•076 533 
.077 583 
.078 631 
.079 676 
1.080 719 
.081 759 
.082 797 
.083 832 
.084 864 
1.085 8 94 



20.00 

19-95 
19.88 
19.85 
19.78 
19.72 
19.68 
19.62 

19-57 
19.52 

19.47 
19-40 
19-35 
19.32 
19.25 
19.20 

19-15 
19.08 
19.05 
19.00 

18.95 
18.88 
18.85 
18.80 
18.75 
18.68 
18.65 
18.60 
18.57 
18.50 

18.45 
18.42 

18-35 
18.3c 
18.27 
18.22 
18.18 

18.13 
18.08 
18.03 
17.98 

17-95 
17.90 
17.85 
17.82 
17.77 

17-73 

17.67 

17-63 
17.60 

17-55 
17-5° 
17-47 

17.42 

17-38 

17-33 
i7-3o 
17-25 
17.20 
17.17 

D. 1". 



9.997 614 
.997 601 
.997 588 
•997 574 
•997 561 

9-997 547 
-997 534 
•997 5 2 ° 
•997 5°7 
•997 493 

9.997 480 
•997 466 
•997 45 2 
•997 439 
•997 425 

9.997 41 1 
•997 397 
•997 383 
•997 369 
•997 355 

9-997 34i 
•997 327 
•997 313 
•997 299 
•997 285 

9.997 271 
•997 257 
•997 242 
.997 228 

•997 214 
9.997 199 
•997 l8 5 
•997 170 
•997 1 5 6 
•997 J 4i 
9.997 127 
.997112 
.997 098 

•997 083 
.997 068 

9-997 053 
•997 °39 
•997 024 
.997 009 
.996 994 

9.996 979 
.996 964 
•996 949 
•996 934 
.996919 

9.996 904 
.996 889 
.996 874 
•996 858 
•996 843 

9.996 828 
.996812 

•996 797 
.996 782 
.996 766 

9-996 75 1 

Sin. 



.22 
.22 

•23 
.22 

•23 

.22 

.23 
.22 

•23 
.22 

.23 

•23 

.22 

•23 
•23 
.23 
•23 
•23 
•23 
•23 
•23 
•23 
.23 
.23 
•23 
•23 
•25 
•23 
•23 
•25 
•23 
•25 
•23 
•25 
•23 

•25 
•23 
•25 
•25 
•25 
•23 
•25 
•25 
•25 
.25 
•25 
•25 
•25 
•25 
•25 
•25 
•25 
.27 

•25 
•25 
.27 
•25 
•25 
•27 
•25 

D. 1". 

83° 



'9.021 620 
.022 834 
.024 044 
.025 251 
•026455 

9.027 655 
.028 852 
.030 046 
.031 237 
.032 425 

9.033 609 

•034 791 
.035 969 

•037 J 44 
.038316 

9.039 485 
.040 65 1 
.041 813 
•042 973 
•044 130 

9.045 284 
•046 434 
.047 582 
.048 727 
.049 869 

9.051 008 
•052 144 

•053 277 
.054 407 

•055 535 

9.056 659 
.057 781 
.058 900 
.060016 
.061 130 

9.062 240 
•063 348 
.064453 
.065 556 
.066 655 

9.067 752 
.068 846 
.069 938 
.071 027 
.072 113 

9.073 197 
.074 278 

•075 356 
.076432 

•077 5°5 

9.078576 

.079 644 

.080 710 

.081 773 

.082 833 

9.083 891 

.084 947 

.086 000 

.087 050 

.088 098 

9.089 144 

Cot. 



20.23 
20.17 
20.12 
20.07 
20.00 

19-95 
19.90 
19.85 
19.80 
19-73 
19.70 
19.63 
19.58 

19-53 
19.48 

19-43 
19-37 
19-33 
19.28 
19.23 
19.17 

I9-I3 
19.08 
19.03 
18.98 

18.93 
18.88 
18.83 
18.80 
18.73 
18.70 
18.65 
18.60 
18.57 
18.50 

18.47 
18.42 
18.38 
18.32 
18.28 
18.23 
18.20 
18.15 
18.10 
18.07 
18.02 
17.97 

17-93 
17.88 
17.85 
17.80 
17.77 
17.72 
17.67 
17-63 
17.60 
17-55 
17-5° 
17-47 
1743 

D. 1". 



0.978 380 
.977 166 
•975 956 
•974 749 
•973 545 

0.972 345 
.971 148 

•969 954 
.968 763 

•967 575 
0.966 391 
.965 209 
.964 03 1 
.962 856 
.961 684 
0.960515 
•959 349 
.958 187 
.957027 

•955 870 
0.954 716 
■953 566 
.952418 
•95i 273 
■95° I3 1 
0.948 992 
•947 856 
.946 723 

•945 593 
•944 465 

0-943 34i 
.942 219 
.941 100 
•939 984 
.938 870 

0.937 760 
.936652 
•935 547 
•934 444 
•933 345 

0.932 248 

•93i 154 
.930 062 
.928 973 
.927887 

0.926 803 
.925 722 
.924 644 
.923 568 
•922 495 

0.921 424 
.920 356 
.919 290 
.918 227 
.917167 

0.916 109 

.915053 
.914000 
.912950 
.911 902 
0.910856 
Tan. 



LOGARITHMIC SINES, COSINES, TANGENTS, 

7° 



AND COTANGENTS. 187 



o 

i 

2 

3 
4 

5 
6 

7 
8 

9 
io 
ii 

12 

13 
14 

15 
16 

17 
18 

19 
20 

21 
22 
23 

24 

25 
26 

27 

28 
29 

30 
3i 
32 
33 
34 

35 
36 
37 
38 
39 
40 

4i 
42 

43 
44 

45 
46 

47 
48 

49 
50 
5i 
52 

53 

54 

55 
56 
57 
58 
59 
60 



Sin. 



9.085 894 
.086 922 
.087 947 
.088 970 
.089 990 

9.091 008 
.092 024 

•093 037 
.094 047 
.095 056 

9.096 062 
.097 065 
.098 066 
.099 065 
.100062 

9.101 056 
.102048 

•103037 
.104025 
.105 010 

9.105 992 
.106973 
.107951 
.108927 
.109 901 

9.110873 
.111 842 
.112 809 
•113 774 
•"4 737 

9.1 15 698 
.116656 
.117 613 
.118567 
.119519 

9.120469 
.121417 
.122 362 
.123306 
.124 248 

9.125 187 
.126 125 
.127060 
.127993 
.128925 

9.129854 
.130781 
.131 706 
.132630 
•133 55i 

9.134470 
•135 387 
• 136303 
.137216 
.138128 

9-139037 
•139 944 
.140 850 

•141 754 
.142655 

9-H3 555 
Cos. 



D. 1". 



I7-I3 
17.08 

17-05 
17.00 
16.97 
16.93 
16.88 
16.83 
16.82 
16.77 
16.72 
16.68 
16.65 
16.62 
16.57 

16.53 
16.48 
16.47 
16.42 
16.37 

16.35 
16.30 
16.27 
16.23 
16.20 
16.15 
16.12 
16.08 
16.05 
16.02 

15-97 
r 5-95 
15.90 
15.87 
15-83 
15.80 
15-75 
15-73 
i5-7o 
15.65 

15.63 

15.58 

15.55 

15-53 
15.48 

15-45 
15.42 
15.40 

15.35 
15.32 
15.28 

15-27 
15.22 
15.20 
15-15 
15.12 
15.10 

15-07 
15.02 
15.00 

D. 1". 



Cos, 



9.996751 

•996 735 
.996 720 

•996 7°4 
.996 688 

9.996 673 
•996657 
.996 641 
.996 625 
.996610 

9.996 594 
•996 578 
.996 562 
•996 546 
•996 530 

9.996514 
.996498 
.996 482 
.996 465 
.996 449 

9-996 433 
.996417 
.996 400 
.996 384 
.996 368 

9.996351 

•996 335 
.996318 
.996 302 
.996 285 

9.996 269 
.996 252 
.996 235 
.996 219 
.996 202 

9.996 185 
.996 168 
.996151 
•996 134 
.996117 

9.996 100 
.996 083 
.996 066 
.996 049 
.996 032 

9.996015 
•995 998 
•995 98o 
•995 963 
•995 946 

9.995 928 
.995911 
.995 894 
•995 876 
•995 859 

9.995 841 
•995 823 
•995 806 
.995 788 
•995 77i 

9-995 753 
Sin. 



D. 1". 



.27 

• 2 5 
.27 
.27 
•25 
.27 
.27 
.27 

• 2 5 
.27 

.27 
.27 
.27 
.27 
.27 
.27 

•27 
.28 
.27 
.27 
.27 
.28 
.27 
.27 
.28 
.27 
.28 
.27 
.28 
.27 
.28 
.28 
.27 
.28 
.28 
.28 
.28 
.28 
.28 
.28 
.28 
.28 
.28 
.28 
.28 
.28 
•30 
.28 
.28 
•30 
.28 
.28 
•30 
.28 
•30 

•30 
.28 
•30 
.28 
•30 



D. 1". 

82^ 



Tan. 



9.089 144 
.090 187 
.091 228 
.092 266 
•093 3°2 

9-094 336 

■095 367 
.096 395 
.097 422 
.098 446 

9.099 468 
.100487 
.101 504 
.102 519 
.103 532 

9.104542 

•i°5 550 
.106 556 

• io 7 559 

.108 560 

9-109 559 
.110556 

111 55i 

•112 543 

■"3 533 

9.114521 

• II 5 5°7 
.116491 
.117472 
.118452 

9-119429 
.120404 
.121377 
.122348 
•123 317 

9.124284 
.125 249 
.126 211 
.127 172 
.128 130 

9.129087 
.130041 
.130994 

•I3 1 944 
.132893 

9-I33 839 

.134784 
.135 726 
.136 667 
.137605 

9.138542 
.139476 
.140409 
.141 340 
.142 269 

9.143 196 
.144121 
.145 044 
.145 966 
.146885 

9-i47 8o3 

Cot. 



D. 1". 



17.38 
17-35 
I7-30 
17.27 

17.23 
17.18 

17.13 
17.12 

17.07 
17-03 
16.98 
16.95 
16.92 
16.88 
16.83 
16.80 
16.77 
16.72 
16.68 
16.65 
16.62 
16.58 
16.53 
16.50 
16.47 
16.43 
16.40 
16.35 
16.33 
16.28 

16.25 
16.22 
16.18 
16.15 
16.12 
16.08 
16.03 
16.02 
J 5-97 
15-95 
15.90 
15.88 

15-83 
15.82 

1577 

15-75 
i5-7o 
15.68 

15-63 
15.62 

r 5-57 
15-55 
I 5-5 2 
15.48 

15-45 
15.42 
I5-38 
J 5-37 
I5-32 
i5-3o 

D. 1". 



Cot. 



0.910 856 
.909813 
.908 772 

•907 734 
.906 698 

0.905 664 

•904 633 
.903 605 
.902578 
•901 554 
0.900 532 

•899 5 X 3 
.898 496 
.897481 
.896 468 

0.895 458 

•894 45° 
.893444 
.892441 
.891 440 
0.890 441 
.889 444 
.888 449 

.887 457 
.886 467 

0.885 479 
.884 493 
.883 509 
.882 528 
.881 548 

0.880 571 
.879 596 
.878 623 
•877652 
.876683 

0.875 7 l6 
.874751 

•873 789 
.872828 
.871 870 

0.870 913 
.869 959 
.869 006 
.868 056 
.867 107 

0.866 161 
.865 216 
.864 274 
•863 333 
.862 395 

0.861 458 
.860 524 

•859 591 
.858 660 

•857 731 
0.856 804 
.855 879 
•854 956 
•854034 
•853 "5 
0.852 197 

Tan. 



60 
59 
58 
57 
56 

55 
54 
53 
52 
5i 
50 
49 
48 
47 
46 

45 
44 
43 
42 

4i 
40 
39 
38 
37 
36 

35 
34 
33 
32 
3i 
30 
29 
28 

27 
26 

25 
24 
23 
22 
21 



19 
18 

17 
16 

15 
14 
13 
12 
11 
10 

9 

8 

7 
6 

5 
4 
3 
2 
I 



188 LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 

8° 



o 

I 

2 

3 
4 

5 
6 

7 
8 

9 

10 

ii 

12 

13 
14 

15 
16 

17 
18 

19 
20 
21 
22 
23 

24 

25 

26 

27 
28 

29 

30 
31 
32 

33 
34 

35 
36 
37 
38 
39 
40 

4i 
42 

43 
44 

45 
46 

47 
48 

49 
50 
5i 
52 
53 
54 

55 

56 
57 
58 
59 
60 



Sin. 



H3 555 
H4 453 

145 349 

146 243 

147 136 
148026 

148 915 

149 802 
150686 
i5 J 5 6 9 

I5245 1 

153 330 

154 208 

155 o8 3 
155 957 
156830 
157 700 
i58 5 6 9 

159 435 

160 301 

161 164 
162025 

162 885 

163743 
164600 

165 454 
166307 
167 159 
168008 
168856 
169 702 
170547 

171 389 

172 230 
173070 

1 73 908 
174744 

175 578 

176 411 
I77 2 42 
178072 
1 78 900 
179726 

180 551 

181 374 

182 196 

183 016 

183834 

184 651 
185466 

186 280 

187 092 

187 903 

188 712 

189 519 

190 325 

191 130 

191 933 

192 734 

193 534 
*94 332 

Cos, 



D.l' 



14.97 
14-93 
14-90 
14.88 
14.83 
14.82 
14.78 
14-73 
14.72 
14.70 
14.65 
14.63 
14.58 
14-57 
H-SS 
14.50 
14.48 
14-43 
14-43 
14.38 

14-35 
H-33 
14.30 
14.28 
14.23 
14.22 
14.20 

14-15 
14-13 
14.10 

14.08 
14.03 
14.02 
14.00 
13-97 

13-93 
13.90 
13.88 
I3-85 
I3-83 
13.80 
13-77 
13-75 
13-72 
13-7° 
13-67 
13-63 
13-62 
13-58 
13-57 
13-53 
I3-5 2 
13,48 

13-45 
1343 
13.42 
13.38 
13-35 
13-33 
13-30 

D. 1". 



Cos. 



9-995 753 
•995 735 
•995 7 J 7 
•995 699 
•995 681 

9.995 664 

•995 646 
.995 628 
.995 610 
•995 59i 

9-995 573 
•995 555 
•995 537 
•995 5 J 9 
•995 5 QI 

9.995 482 
•995 464 
•995 446 
•995 427 
•995 409 

9-995 39o 
•995 372 
•995 353 
•995 334 
•995 3i6 

9-995 297 
•995 278 
.995 260 

•995 241 
-995 22 2 

9-995 203 
•995 l8 4 
•995 l6 5 
•995 J 46 
•995 I2 7 

9.995 108 
•995 089 
•995 070 
•995 05 1 
•995 032 

9-995 OI 3 
•994 993 
•994 974 
•994 955 
•994 935 

9.994916 
.994 896 

•994 877 
•994 857 
•994 838 

9.994818 
•994 798 
•994 779 
•994 759 
•994 739 

9.994 720 

•994 7°o 
.994 680 
.994 660 
.994 640 
9.994 620 
Sin. 



D. 1" 



D. 1". 

~8P 



Tan. 



9.147803 
.148718 
.149632 
.150 544 
•I5 r 454 

9.152363 
.153269 
.154174 
•155 077 
•155 978 

9.156877 

•157 775 
.158671 

•159 565 
.160457 

9.161 347 
.162 236 
.163123 
.164008 
.164892 

9.165 774 
.166 654 

.167532 
.168409 
. 1 69 284 

9.170 157 
.171 029 
.171 899 
.172 767 
•173 634 

9.174499 
.175362 
.176 224 
.177084 
.177942 

9.178799 

•179655 
.180508 
.181 360 
.182 211 
9.183059 
.183907 
.184752 

•185 597 
.186439 

9.187 280 
.188 120 
.188958 
.189 794 
.190629 

9.191 462 
.192294 
.193124 

•193 953 
.194 780 

9.195 606 
.196430 

.197253 
.198074 
.198 894 

9-I99 7I3 

Cot. 



D. 1". 



I5-25 
I5-23 
15.20 

15.17 
15.15 

15.10 
15.08 

15-05 
15.02 

14.98 

14.97 

14.93 
14.90 
14.87 
14.83 
14.82 
14.78 
H.75 
1473 
14.70 

14.67 
14.63 
14.62 
14.58 
H-55 

14.53 
14.50 
14.47 

1445 
14.42 

14.38 
14.37 
14-33 
14.30 
14.28 
14.27 
14.22 
14.20 
14.18 
I4-I3 

14-13 

14.08 
14.08 
14.03 
14.02 
14.00 
13-97 
13-93 
13.92 
13-88 

13-87 
13-83 
13.82 

13-78 
13-77 
13.73 
13-72 
13.68 

13-67 
13-65 

D. 1". 



Cot. 



0.852 197 
.851 282 
.850 368 
.849 456 
.848 546 

0.847 637 
.846 731 
.845 826 
.844 923 
.844 022 

0.843 123 
.842 225 
.841 329 
.840 435 
•839 543 

0.838 653 
.837 764 
.836877 
.835 992 
.835 108 

0.834 226 

•833 346 
.832 468 
.831 591 
.830 716 

0.829 843 
.828971 
.828 101 
.827 233 
.826 366 

0.825 501 
.824 638 
.823 776 
.822 916 
.822058 

0.821 201 
.820 345 
.819492 
.818640 
.817789 

0.816 941 
.816093 
.815248 
.814403 
.813561 

0.812 720 
.811 880 
.811 042 
.810 206 
.809371 

0.808 538 
.807 706 
.806 876 
.806 047 
.805 220 

0.804 394 
.803 570 
.802 747 
.801 926 
.801 106 

0.800 287 
Tan. 



LOGARITHMIC SINES, CO SIXES, TANGENTS, AND COTANGENTS. 189 



Sin. 



9-194 332 
.195 129 

•195 925 
.196 719 

.197 5 11 

9.198 302 
.199091 
.199879 
.200 666 
.201 451 

9.202 234 
.203 017 
.203 797 
.204577 
.205 354 

9.206 131 
.206 906 
.207 679 
.208 452 
.209 222 

9.209 992 
.210 760 
.211 526 
.212 291 
•2i3 55 

9.213 818 
•214 579 
•215 338 
.216097 
.216 854 

9.217 609 
.218363 
.219 116 
.219868 
.220 618 

9.221 367 
.222 115 
.222 861 
.223 606 
•224 349 

9.225 092 
.225 833 
.226573 
.227311 
.228 048 

9.228 784 
.229 518 
.230 252 
.230 984 
•231 715 

9.232 444 
•233 172 
•233 899 
•234 625 
•235 349 
9.236073 
.236 795 

•237 5 J 5 
.238 235 

•238953 

9-239 670 

Cos. 



D. 1". 



13.28 
13-27 
13-23 
13.20 
13.18 

13-15 
I3-I3 
13.12 
13.08 
13.05 

13-05 
13.00 
13.00 
12.95 
12.95 
12.92 
12.88 
12.88 
12.83 
12.83 
12.80 
12.77 
12.75 
12.73 
12.72 

12.68 
12.65 
12.65 
12.62 
12.58 

12.57 
12-55 
12.53 
12.50 
12.48 
12.47 
12.43 
12.42 
12.38 
12.38 

12-35 
12-33 
12.30 
12.28 
12.27 
12.23 
12.23 
12.20 
12.18 
12.15 
12.13 
12.12 
12.10 
12.07 
12.07 
12.03 
12.00 
12.00 
11.97 
"•95 

D. 1". 



Cos. 



9.994 620 
.994 600 

•994 5 8 ° 
.994 560 

•994 540 

9-994 5 l 9 
•994 499 
•994 479 
•994 459 
•994 438 

9.994418 
•994 398 
•994 377 
•994 357 
•994 336 

9.994316 
•994 295 
•994 274 
•994 254 
•994 233 

9.994212 

•994 191 
.994171 

-994 15° 
.994129 

9.994 108 
•994 087 
.994 066 
•994 045 
•994 024 

9.994 003 
•993 982 
•993 96o 
•993 939 
.993918 

9-993 897 
•993 875 
•993 854 
•993 832 
.993811 

9-993 789 
•993 7 68 
•993 746 
•993 725 
-993 7°3 

9.993 681 
•993 660 
•993 638 
•993 616 
•993 594 

9-993 572 
•993 55° 
•993 528 
•993 5° 6 
-993 484 

9.993 462 

•993 440 
.993418 
•993 396 
•993 374 
9-993 35 1 
Sin. 



D. 1". 



•oj 
-33 
•33 
'33 
•35 
-33 
■33 
'33 
•35 
•33 
-33 
-35 
-33 
•35 
-33 
•35 
•35 
■33 
•35 
•35 
•35 
■33 
•35 
•35 
•35 
•35 
•35 
•35 
•35 
■35 
•35 
-37 
•35 
•35 
•35 
•37 
■35 
-37 
■35 
■37 
•35 
•37 
•35 
•37 
■37 
■35 
■37 
•37 
■37 
-37 
•37 
-37 
•37 
■37 
•37 
•37 
-37 
•37 
•37 
.38 

D. 1" 

80° 



Tan. 



9-199 7 J 3 
.200 529 
.201 345 
.202 159 
.202 971 

9.203 782 
.204 592 
.205 400 
.206 207 
.207013 

9.207 817 
.208 619 
.209 420 
.210 220 
.211 018 

9. 211 815 
.212 611 
.213405 
.214 198 
.214989 

9.215 780 
.216568 

•217 35 6 
.218 142 
.218 926 

9.219 710 
.220 492 
.221 272 
.222052 
.222 830 

9.223 607 
.224382 
.225 156 
.225 929 
.226 700 

9.227471 
.228 239 
.229007 
-229 773 
•230 539 

9.231 302 
.232 065 
.232 826 
-233 586 
•234 345 

9.235 103 

•235 859 
.236614 

.237 368 
.238 120 

9.238872 
.239 622 
.240 371 
.241 118 
.241 865 

9.242 610 
•243 354 
.244097 
.244 839 

•245 579 

9-246 3^9 

Cot. 



D. 1' 



3.60 
3.60 
3-57 
3-53 

3-5 2 

3-5° 
3-47 
3-45 
3-43 
3-4o 

3-37 
3-35 
3-33 
3-30 
3-28 

3-27 
3.23 
3.22 

3-i8 
3-i8 

3-13 
3-*3 
3.10 

3-07 
3-°7 

3-03 
3.00 
3.00 
2-97 
2-95 
2.92 
2.90 
2.88 
2.85 
2.85 
2.80 
2.80 
2.77 
2.77 
2.72 
2.72 
2.68 
2.67 
2.65 
2.63 
2.60 
2.58 
2-57 
2-53 
2-53 
2.50 
2.48 
2-45 
2-45 
2.42 

2.40 
2.38 
2-37 
2-33 
2-33 

B. 1". 



Cot. 



0.800 287 

•799 47 1 
•798655 

-797 841 
.797029 

0.796 218 
•795 408 
.794 600 

•793 793 
•792987 
0.792 183 
.791 381 
.790 580 
.789 780 
788 982 
788 185 
787 389 
786 595 
785 802 
785 01 1 
784 220 

783 432 
782 644 
781 858 
781 074 
780 290 

779 5°8 
778 728 

777 948 
777170 

77 6 393 
775618 

774 844 
774071 
773 300 

772529 
771 761 

770 993 

770 227 
769 461 
768 698 

7 6 7 935 
767 174 
766414 
765 655 

764 897 
764 141 
763 386 
762 632 
761 880 
761 128 
760 378 
759629 
758 882 
758 135 

757 39o 
756 646 

755 903 
755 161 
754 421 
753 68 i 

Tan. 



60 
59 
58 
57 
56 

55 

54 
53 
53 
5i 
50 
49 
48 

47 
46 

45 
44 
43 
42 

4i 
40 
39 
38 
37 
36 

35 

34 
33 
32 
3i 
30 
29 
28 
27 
26 

25 
24 

23 
22 

21 

20 

19 
18 

17 
16 

15 
14 
13 
12 
11 
10 

9 
8 

7 
6 

5 
4 
3 
2 

1 



190 LOGARITHMIC SINES, COSINES, TANGENTS, 

10° 



AND COTANGENTS. 



M. 


Sin. 


D. 1". 


Cos, 


D. 1". 


Tan. 


D. 1", 


Cot. 




o 


9.239 670 


H-93 
11.92 
11.88 
11.87 
11.85 

n.83 
11.82 


9-993 35 1 


■37 
-37 
■38 
■37 
■37 


9.246319 




12.30 

12.28 


753 681 


60 


i 


.240 386 


•993 329 


.247057 


752 943 


59 


2 


.241 101 


•993 3°7 


.247 794 


12.27 
12.23 
12.23 


752 206 


58 


3 


.241 814 


•993 284 


.248 530 


75M70 


57 


4 


.242 526 


.993 262 


.249 264 


75° 736 


56 


5 


9-243 2 37 


9-993 240 


■3* 
•37 
.38 
?8 


9.249 998 


3 
12.20 


750 002 


55 


6 


.243 947 


.993217 


.250 730 


12.18 


749 270 


54 


7 


.244 656 


11.78 
11.77 
11.77 
11.72 
11.72 


•993 195 


.251 461 


12.17 
12.15 

12.13 


12.10 

1 2.10 


748 539 


53 


8 


•245 363 


.993172 


.252 191 


747 809 


52 


9 


.246 069 


•993 H9 


•37 
•38 
^8 


.252920 


747 080 


5i 


IO 

ii 


9.246 775 
.247 478 


9.993 127 
•993 104 


9.253 648 

•254 374 


746 352 

745 626 


50 
49 


12 


.248 181 


•993 081 


•o° 


.255 100 




744 9oo 


48 


13 


.248 883 


11.70 

11.67 
11.65 


•993 059 


•37 
■3^ 
•3% 


.255 824 


1 2.07 
12.05 
12.03 


744176 


47 


14 


.249 583 


•993 36 


.256547 


743 453 


46 


15 


9.250 282 


11.63 
11.62 


9-993 0I3 


.38 
•38 
38 


9.257 269 



12.02 


742 731 


45 


16 


.250 980 


.992 990 


.257990 


1 2. CO 


742 010 


44 


17 


.251677 


1 1.60 


.992967 


.258 710 


11.98 
n-95 
11.95 
11.92 
11.90 
11 88 


741 290 


43 


18 


•252 373 


n-57 
n-57 

11-53 
11.52 

11.50 
11.48 
11.47 

11.45 
11.42 

11.42 

n.38 

11-37 

n-35 


•992 944 


•38 
•38 


•259429 


740571 


42 


19 


.253067 


.992 921 


.260 146 


739 854 


4i 


20 


9.253 761 


9.992 898 


■38 
•38 
•38 
.38 
•3^ 


9.260 863 


739 137 


40 


21 


.254453 


.992875 


.261 578 


738422 


39 


22 


•255 J 44 


.992852 


.262 292 


737 7o8 


38 


23 


•255 834 


.992 829 


.263 005 


n'.8 7 
11.85 


736 995 


37 


24 


.256523 


.992 806 


.263717 


736 283 


36 


25 


9.257 211 


9.992 783 


.40 
•3^ 
•38 
•38 
.40 

•38 
.40 

.38 
.40 

•3^ 


9.264428 


...83 ° 

11.82 


735 572 


35 


26 


.257 898 


.992 759 


.265 138 


734 862 


34 


27 


.258 583 


.992 736 


.265 847 


II.SO 


.734 153 


33 


28 


.259 268 


.992713 


.266555 


11.77 

11.77 

n-73 
n-73 
11.70 
11 70 


733 445 


32 


29 


.259951 


.992 690 


.267 261 


732 739 


3i 


30 


9.260 633 


9.992 666 


9.267 967 


732 033 


30 


31 


.261 314 


•992 643 


.268 671 


73i 329 


29 


32 


.261 994 


1I 33 
11.32 
11.30 
11.27 
11.27 
11.23 
11.23 
11.20 
11.20 


.992 619 


•269 375 


730 625 


28 


33 


.262 673 


.992 596 


.270077 


729 923 


27 


34 


•263351 


.992572 


.270 779 


11.67 


729 221 


26 


35 


9.264027 


9.992 549 


.40 
.40 

.38 
.40 
.40 
.40 


9.271 479 


11.65 ° 

11-63 

11.62 


728521 


25 


36 


.264 703 


.992 525 


.272 178 


727 822 


24 


37 


.265 377 


.992 501 


.272 876 


727 124 


23 


38 
39 


.266051 
.266 723 


•992 478 
.992 454 


.273 573 
.274 269 


11.60 
11.58 

n-57 
"■55 
n-53 
11.52 
11.50 


726427 
.725 73i 


22 
21 


40 


9.267 395 


11. 17 


9.992 430 


9.274 964 


725 036 


20 


4i 


.268065 


.992 406 


.275 658 


724 342 


19 


42 


.268 734 


11. 15 
11. 13 
1 1 .12 


.992 382 


.40 

.38 
.40 
.40 


.276351 


723 649 


18 


43 


.269 402 


•992 359 


.277043 


722957 


17 


44 


.270069 


11. 10 


•992 335 


•277 734 


722 266 


16 


45 
46 


9.270 735 
.271 400 


11.08 
11.07 
11.03 


9.992 31 1 
.992 287 


.40 
.40 
.40 


9.278 424 
.279113 


11.48 ° 

11.47 

"■45 

"•43 

11.40 


721 576 
.720 887 


15 
14 


47 


.272 064 


•992 263 


.279 801 


720 199 


13 


48 


.272 726 


.992 239 


.280 488 


7 I 95 12 


12 


49 


.273 388 


11.03 

11.02 


.992 214 


.42 
.40 


.281 174 


718826 


11 


50 


9.274049 


10.98 
10.98 
10.97 


9.992 190 


.40 
.40 
.40 


9.281 858 




11.40 

".38 
"•37 
"•35 
11.33 


718 142 


10 


5i 


.274 708 


.992 166 


.282 542 


717458 


9 


52 


.275 367 


.992 142 


.283 225 


716775 


8 


53 


.276025 


.992 118 


.283 907 


716093 


7 


54 


.276681 


10.93 
10.93 
10.90 


.992 093 


.42 
.40 

•42 
.40 
.40 
.42 
.40 


.284 588 


715 4i2 


6 


55 


9-277 337 


9.992 069 


9.285 268 


^ 
11.32 

11.28 


7H732 


5 


56 


.277991 


•992 044 


•285 947 


7H°53 


4 


57 


.278 645 


10.90 
10.87 
10.85 
10.85 


.992 020 


.286 624 


11.28 


7*3 376 


3 


58 


.279 297 


.991 996 


.287 301 


11.27 
11.25 


712699 


2 


59 


.279948 


.991971. 


.287 977 


712023 


1 


60 


9.280 599 


9.991 947 




9.288652 


D 


711 348 







Cos. 


D. 1". 


Sin. 


D. 1". 


Cot. 


D. 1". 


Tan. 


M. 



79° 



LOGARITHMIC SIXES, COSINES, TANGENTS, AND COTANGENTS. 191 

11° 



M. 



Sin. 



D, 1". 



Cos, 



D. 1". 



Tan, 



D. 1' 



Cot. 



o 

i 

2 

3 
4 

5 
6 

7 
8 

9 

10 

ii 

12 

13 

14 

15 
16 

17 
18 

19 
20 
21 
22 

23 
24 

25 
26 
27 
28 
29 
30 
3i 
32 
33 
34 

35 
36 
37 
38 
39 
40 
4i 
42 
43 
44 

45 
46 

47 
48 

49 
50 
51 
52 
53 
54 

55 
56 
57 
58 
59 
60 



9.280 599 
.281 248 
.281 897 
.282 544 
.283 190 

9.283 836 
.284480 
.285 124 
.28c; 766 
.286 408 

9.287 048 
.287 688 
.288 326 
.288 964 
.289 600 

9.290 236 
.290 870 
.291 504 
.292 137 
.292 768 

9-293 399 
.294 029 
.294 658 
.295 286 
.295913 

9.296 539 
.297 164 
.297 788 
.298412 
.299 034 

9-299 655 
.300 276 
.300 895 

•3 01 5*4 
.302 132 

9.302 748 
■3°3 364 
•303 979 
.304 593 
.305 207 



9- 



305 819 

306 430 

307 041 

307 650 

308 259 
308 867 

3°9 474 
310080 
310685 
311 289 
311 893 
3^495 
313 097 
313698 

3H297 

3H897 
315 495 
316092 
316689 
317284 

317 879 
Cos. 



0.82 
0.82 
0.78 

0-77 
0.77 

o.73 
0.73 
0.70 
0.70 
0.67 
0.67 
0.63 
0.63 
0.60 
0.60 

o-57 
o.57 
°-55 
0.52 
0.52 
0.50 
0.48 
0.47 
0.45 
0.43 
0.42 
0.40 
0.40 
o-37 
o.35 

o.35 
0.32 
0.32 
0.30 
0.27 
0.27 
0.25 
0.23 
0.23 
0.20 
0.18 
0.18 
0.15 
0.15 
0.13 
0.12 
0.10 

0.08 

0.07 
0.07 

0.03 

0.03 
0.02 

9.98 

10.00 

9-97 
9-95 
9-95 
9.92 
9.92 

D. 1". 



9.991 947 
.991 922 
.991 897 
.991 873 
.991 848 

9.991 823 
.991 799 
.991 774 
.991 749 
.991 724 

9.991 699 
.991 674 
.991 649 
.991 624 
•99i 599 

9-991 574 
•991 549 
.991 524 
.991 498 
•99i 473 

9.991 448 
.991 422 

•99i 397 
.991 372 
.991 346 

9.991 321 
.991 295 
.991 270 
.991 244 
.991 218 

9.991 193 
.991 167 
.991 141 
.991 115 
.991 090 

9.991 064 
.991 038 
.991 012 
.990 986 
.990 960 

9-990 934 
.990 908 
.990 882 
•990855 
.990 829 

9.990 803 
•99o 777 
•990 75° 
•99o 724 
•990 697 

9.990 671 
.990 645 
.990 618 
.990 591 
•990 5 6 5 

9-99o 538 
.990511 
•990 485 
.990 458 
.990431 

9.990 404 
Sin. 



.42 
.42 
.40 
.42 
.42 
.40 
.42 
.42 
.42 
.42 
.42 
.42 
.42 
.42 
.42 
.42 
.42 

•43 
.42 
.42 

•43 
.42 
.42 

•43 
.42 

•43 
.42 

•43 
•43 
.42 

•43 
•43 
•43 
.42 

•43 
•43 
•43 
•43 
•43 
•43 
•43 
•43 
•45 
•43 
•43 
•43 
•45 
•43 
•45 
•43 
•43 
•45 
•45 
•43 
•45 
•45 
•43 
•45 
•45 
•45 

P. 1", 

~78° 



9.288652 
.289 326 
.289 999 
.290 671 
.291 342 

9.292013 
.292 682 

.293 350 
.294017 
.294 684 

9-295 349 
.296013 
.296 677 

•297 339 
.298 001 

9.298 662 
.299 322 
.299 980 
.300 638 
.301 295 

9-301 951 

.302 607 
.303 261 
•303 9H 
.304 5 6 7 

9.305 218 
.305 869 
.306519 
.307 168 
.307816 

9.308 463 
.309 109 
•309 754 
•3!0 399 
.311042 

9.31 1 685 
.312327 
.312 968 
.313608 
.314 247 

9.314885 
.3i5 5 2 3 
•316 159 
•316795 
•3i7 43o 

9.318064 
.318697 

•3*9 33° 
.319961 
.320 592 

9.321 222 
•321 851 
.322479 
.323 106 
•323 733 

9-324 358 
.324 983 
.325607 
.326231 
.326853 

9.327 475 
Cot. 



1.23 
1.22 
1.20 
1. 18 
1. 18 

115 

i-i3 
1.12 
1. 12 

1.08 
1.07 
1.07 
1.03 
1.03 
1.02 
1. 00 
0.97 
0.97 
o.95 
o-93 

o-93 
0.90 
0.88 
0.88 
0.85 
0.85 
0.83 
0.82 
0.80 
0.78 
0.77 
o.75 
o-75 
0.72 
0.72 
0.70 
0.68 
0.67 
0.65 
0.63 
0.63 
0.60 
0.60 
0.58 
0-57 
o.55 
o-55 
0.52 
0.52 
0.50 
0.48 
o-47 
0.45 
0.45 
0.42 

0.42 
0.40 
0.40 
o-37 
o.37 

D. 1", 



0.71 1 348 
.710674 
.710001 
.709 329 
.708 658 

0.707 987 
.707318 
.706 650 

•705 983 
.705316 

0.704 651 
•7°3 987 
•703 323 
.702 661 
.701 999 

0.701 338 
.700 678 
.700020 
•699 362 
.698 705 

0.698 049 

•697 393 
.696 739 
.696086 
•695 433 
0.694 782 
.694 131 
.693 481 
.692 832 
.692 184 

0.691 537 
.690 891 
.690 246 
.689 601 
.688 958 

0.688315 
.687 673 
.687 032 
.686 392 
•685 753 
, 0.685 IX 5 
.684477 
.683 841 
.683 
.682 



205 



57o 
0.681 936 



.681 
.6806 
.680 
.679 

0.678 
.678 
.677 
.676 
.676 

0.675 

•675 
.674 



303 
670 

039 

408 



■673 

.673 

0.672 



778 
149 

521 
894 
267 

642 
017 

393 
769 

H7 

525 



60 
59 
58 
57 
56 

55 
54 
53 
52 
5i 
50 
49 
48 
47 
46 

45 
44 
43 
42 
4i 
40 
39 
38 
37 
36 

35 
34 
33 
32 
31 
30 
29 
28 

27 
26 

25 
24 
23 
22 
21 



19 
18 

17 
16 

15 
14 
13 
12 
11 



Tan. 



192 LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 

12° 



M. 


Sin. 


o 


9-3 I 7 879 


i 


.318473 


2 


.319066 


3 


.319658 


4 


.320 249 


5 


9.320 840 


6 


.321 430 


7 


.322019 


8 


.322 607 


9 


•323 194 


IO 


9.323 780 


ii 


•324 3 66 


12 


•32495° 


13 


.325 534 


H 


.326117 


15 


9.326 700 


16 


.327 281 


17 


.327 862 


18 


.328 442 


19 


.329021 


20 


9.3 2 9 599 


21 


.330 176 


22 


•330 753 


23 


•33 1 3 2 9 


24 


.331 903 


25 


9-332 478 


26 


■333 05 1 


27 


-333 624 


28 


•334 195 


29 


•334 767 


30 


9-335 337 


31 


•335 9o6 


32 


.336 475 


33 


•337 °43 


34 


•337 610 


35 


9-338i76 


36 


.338 742 


37 


•339 307 


38 


•339 871 


39 


.340 434 


40 


9.340 996 


4i 


•34i 558 


42 


.342119 


43 


.342 679 


44 


•343 239 


45 


9-343 797 


46 


•344 355 


47 


.344912 


48 


•345 469 


49 


.346 024 


50 


9-346 579 


5i 


•347 134 


52 


•347 687 


53 


.348 240 


54 


.348 792 


55 


9-349 343 


56 


•349 893 


57 


•35° 443 


58 


.350 992 


59 


•35 1 540 


60 


9.352088 



D. 1". 



Cos. 



9.90 
9.88 
9.87 
9.85 
9.85 

9-83 
9.82 
9.80 
9.78 
9-77 
9-77 
9-73 
9-73 
9.72 
9.72 
9.68 
9.68 
9.67 
9-65 
963 
9.62 
9.62 
9.60 
9-57 
9.58 

9-55 
9-55 
9-5 2 
9-53 
9-5o 
9.48 
9.48 
9-47 
9-45 
9-43 

9-43 
9.42 
9.40 
9.38 
9-37 
9-37 
9-35 
9-33 
9-33 
9-3o 

9-30 
9.28 
9.28 
9-25 
9-25 

9-25 
9.22 
9.22 
9.20 
9.18 
9.17 
9.17 
9-15 
9-i3 
9-13 

D. 1". 



Cos. 



9.990 404 
.990 378 
.990351 
.990 324 
.990 297 

9.990 270 
.990 243 
.990 215 
.990 188 
.990 161 

9.990 134 
.990 107 
.990079 
.990052 
.990025 

9.989 997 
.989 970 
.989 942 
.989915 
.989 887 

9.989 860 
.989 832 
.989 804 
.989 777 
•989 749 

9.989 721 

.989 693 
.989 665 
.989 637 
.989 610 

9.989 582 
.989 553 
•989 5 2 5 
.989 497 
.989 469 

9.989 441 
.989413 
.989385 
.989 35 6 
.989 328 

9.989 300 
.989 271 

.989 243 
.989 214 
.989 186 

9-989 157 
.989 128 
.989 100 
.989071 
.989 042 

9.989014 
.988 985 
.988956 
.988 927 
.988 898 

9.988 869 
.988 840 
.988811 
.988 782 
.988 753 

9-988 724 

Sin. 



D. 1". 



•43 
•45 
•45 
•45 
•45 
•45 
•47 
•45 
•45 
•45 
•45 
•47 
•45 
•45 
•47 
•45 
•47 
•45 
•47 
•45 
•47 
•47 
•45 
•47 
•47 
•47 
•47 
•47 
•45 
•47 
.48 
•47 
•47 
•47 
•47 
•47 
•47 
.48 

•47 
•47 

.48 
•47 
.48 
•47 
.48 

.48 

•47 
.48 
.48 
•47 
.48 
.48 
.48 
.48 
.48 

.48 

.48 



.48 

D. 1". 

770 



Tan. 



9-327 475 
.328 095 

•328715 
•329 334 
•3 2 9 953 

9-330 570 
•33i 187 
•331 803 
•332418 
■333 o33 

9-333 646 
•334 259 
•334 871 
•335 482 
•33 6 °93 

9.336 702 
•337 3" 
.337 919 
•338 5 2 7 
-339 133 

9-339 739 
.340 344 
•340 948 
•34i 552 
•342 155 

9-342 757 
•343 358 
•343 958 
•344 558 
•345 x 57 

9-345 755 
•346 353 
.346 949 

•347 545 
•348 141 
9-348 735 
•349 329 
•349 922 
.350 5H 
.351 106 

9.351 697 
.352287 
.352876 
•353 465 
.354 053 

9.354 640 
•355 227 
.355 813 
.35 6 398 
•356982 

9-357 566 
.358 149 
•358 73i 
.359 313 
•359 893 

9.360 474 
361 053 
.361 632 
.362 210 
.362 787 

9-363 364 
Cot. 



D. V 



IO-33 
10.33 
10.32 
10.32 
10.28 
10.28 
10.27 
10.25 
10.25 
10.22 
10.22 
10.20 
10.18 
10.18 
10.15 
10.15 
10.13 
10.13 
10.10 
10.10 
10.08 
10.07 
10.07 
10.05 
10.03 
10.02 
10.00 
10.00 
9.98 
9-97 
9-97 
9-93 
9-93 
9-93 
9.90 

9.90 
9.88 
9.87 
9.87 
9.85 

9-83 
9.82 
9.82 
9.80 
9.78 
9.78 
9-77 
9-75 
9-73 
9-73 
9.72 
9.70 
9.70 
9.67 
9.68 

9-65 
9-65 
9-63 
9.62 
9.62 

D. 1". 



Cot. 



0.672525 
.671 905 
.671 285 
.670 666 
.670 047 

0.669 430 
.668813 
.668 197 
.667 582 
.666 967 

0.666 354 
.665 741 
.665 129 
.664518 
.663 907 

0.663 298 
.662 689 
.662081 
.661 473 
.660 867 

0.660 261 
.659656 
.659052 
.658448 
.657845 

0.657 2 43 
.656 642 
.656 042 

•655 442 
.654 843 

0.654 245 
.653 647 
.653051 
.652455 
.651859 

0.651 265 
.650671 
.650 078 
.649 486 
.648 894 

0.648 303 

.647 7 l 3 
.647 124 

.646 535 

.645 947 

0.645 360 

.644 773 
.644 187 
.643 602 
.643018 

0.642 434 
.641851 
.641 269 
.640 687 
.640 107 

0.639 526 
.638 947 
.638 368 
.637 79o 
•637213 

0.636 636 

Tan. 



LOGARITHMIC SIXES, COSINES, TANGENTS, AND COTANGENTS. 193 

13° 



Sin. 



9.352088 
•35 2 6 35 
•353 181 
•353 726 
.354 271 

9-354 8I5 
•355 358 
•355 901 
•35 6 443 
.356 984 

9-357 5 2 4 
.358064 
.358 603 
•359 141 
•359 678 

9.360 215 
.360 752 
.361 287 
.361 822 
.362 356 

9.362889 
.363 422 
■363 954 
•364 485 
.365016 

9-3 6 5 546 
.366075 
.366 604 
.367 131 
-367 659 

9.368 185 
.368 711 
.369 236 
.369 761 
•370 285 

9.370 808 
-37 1 330 
•37i 852 
•372 373 
.372 894 

9-373 4H 
•373 933 
•374 45 2 
•374 97° 
•375 487 

9.376003 
•37 6 5 J 9 
•377 035 
•377 549 
.378063 

9-378 577 
•379 089 
.379 601 
.380113 
.380 624 

9-38i 134 
.381 643 
.382 152 
.382 661 
.383 168 

9-383 675 
Cos. 



D, V 



9.12 
9.10 
9.08 
9.08 
9.07 

9-05 
9-05 
9-03 
9.02 
9.00 
9.00 
8.98 

8-97 
8.95 
8-95 
8.95 
8.92 
8.92 
8.90 
8.88 
8.88 
8.87 
8.85 
8.85 
8.83 
8.82 
8.82 
8.78 
8.80 
8-77 

8-77 
8.75 

8.75 
8-73 
8.72 

8.70 
8.70 
8.68 
8.68 
8.67 
8.65 
8.65 
8.63 
8.62 
8.60 
8.60 
8.60 
8.57 
8.57 
8.57 

8-53 
8-53 
8-53 
8.52 
8.50 
8.48 
8.48 
8.48 
8-45 
8-45 

D. 1". 



Cos. 



D. 1". 



9.988 724 
.988 695 
.988 666 
.988 636 
.988 607 

9.988578 
.988 548 
.988519 
.988 489 
.988 460 

9.988430 
.988 401 
.988 371 
.988 342 
.988312 

9.988 282 
.988 252 
.988 223 
.988 193 
.988 163 

9.988133 
.988 103 
.988073 
.988 043 
.988013 

9.987983 

.987 953 
.987 922 
.987 892 
.987 862 
9.987 832 
.987 801 

•987 77i 

.987 740 

.987 710 

9.987 679 

•987 &49 
.987618 
.987 588 
•987 557 

9.987 526- 
.987 496 
.987 465 
•987 434 
.987 403 

9-987 372 
.987 34i 
.987 3io 
.987 279 
.987 248 

9.987217 
.987 186 

.987 155 

.987 124 

.987 092 

9.987061 

•987 030 
.986 998 
.986 967 
.986 936 
9.986 904 
Sin, 



.48 
.48 
•50 
.48 
.48 

.48 

•50 

.48 

•5° 
. 4 8 

•50 
.48 
•5° 
•50 

•5° 
.48 

•5° 
•50 
•50 
•5° 
•5° 
•5° 
•50 
•5° 

•50 
•52 
•50 
•5° 
•5° 
•52 
•5° 
•52 
•5° 
•52 

•50 

.52 
•5° 
•52 
•52 

•50 
•52 
•52 
•52 
•52 
•52 
•52 
•52 
•52 
•52 

•5 2 
•52 
•5 2 
•53 
•52 
•52 
•53 
•52 
•5 2 
•53 

D 1" 

~76° 



Tan. 



9-363 364 
•363 940 

•364 5 1 5 
.365 090 

•365 664 

9.366 237 

.366810 

.367 382 

•367 953 

•368 524 

9.369 094 
•369 663 
.370 232 
-37° 799 
•37 1 367 

9-371 933 
•372 499 
•373064 
•373 629 
•374 193 

9-374 756 
•375 319 
•375 881 
•376 442 
•377 003 

9-377 5 6 3 
.378 122 
.378681 
-379 239 
•379 797 

9-38o 354 
.380910 
.381 466 
.382020 
•382 575 

9.383 129 
.383 682 

.384 234 
.384 786 
-385 337 
9.385 888 
.386438 
.386987 

.387 536 
.388 084 

9.388 631 

.389 178 
.389 724 
.390 270 
.390815 
9.391 360 
•39i 903 
•392 447 
.392 989 

•393 531 
9-394 073 
•394 614 
•395 *54 
•395 694 
•396 233 
9-396 771 
Cot. 



D. 1' 



9.60 
9-58 
9-58 
9-57 
9-55 
9-55 
9-53 
9-52 
9-5 2 
9-5° 
948 
9.48 
9-45 
9-47 
9-43 

9-43 
9.42 
9.42 
9.40 
9-38 
9.38 
9-37 
9-35 
9-35 
9-33 
9-32 
9-32 
9-30 
9-30 
9.28 

9.27 
9.27 
9-23 
9-25 
9-23 
9.22 
9.20 
9.20 
9.18 
9.18 
9.17 
9.15 
9-15 
9-13 
9.12 

9.12 
9.10 
9.10 

9.08 
9.08 

9-05 
9.07 

903 
9-03 
903 
9.02 
9.00 
9.00 
8.98 
8.97 

D. 1". 



Cot. 



0.636 636 
.636 060 

.635 485 
.634910 

.634 336 
0.633 763 
•633 J 90 
.632 618 
.632047 
.631 476 
0.630 906 

•630 337 
.629 768 
.629 201 
.628 633 

0.628 067 
.627 501 
.626 936 
.626 371 
.625 807 

0.625 244 
.624681 
.624 119 

.623 558 
.622 997 

0.622 437 
.621 878 
.621 319 
.620 761 
.620 203 

0.619 646 
.619 090 
•618534 
.617 980 
.617425 

0.616 871 
.616318 
.615 766 
.615 214 
.614 663 

0.614 112 
.613 562 
.613013 
.612464 
.611 916 

0.61 1 369 
.610 822 
.610 276 
.609 730 
.609 185 

0.608 640 
.608 097 

•607 553 
.607011 
.606 469 

0.605 927 
.605 386 
.604 846 
.604 306 
.603 767 

0.603 229 
Tan. 



60 
59 
58 
57 
56 

55 
54 
53 
52 
5i 
50 
49 
48 

47 
46 

45 
44 
43 
42 

4i 
40 
39 
38 
37 
35 
35 
34 
33 
32 
3i 
30 
29 
28 
27 
26 

25 
24 

23 
22 
21 



19 
18 

17 
16 

15 
M 
13 
12 
11 
10 

9 
8 

7 
6 

5 
4 
3 
2 

1 



M. 



194 LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 

14° 



M. 



o 

i 

2 

3 
4 

5 
6 

7 
8 

9 

10 

ii 

12 

13 
14 

15 
16 

17 
18 

19 

20 
21 
22 
23 

24 

25 

26 

27 
28 

29 
30 
31 
32 

33 
34 

35 
36 
37 
38 
39 
40 
4i 
42 
43 
44 

45 
46 

47 
48 

49 
5o 
5i 
52 
53 
54 

55 
56 
57 
58 
59 
60 



Sin. 



9- 



■383 675 
.384 182 
.384 687 

.385 l 92 
.385 697 

.386 201 
.386 704 
.387 207 
.387 709 
.388210 

.388711 
.389211 
.389711 
.390 210 
.390 708 
391 206 

•39i 7°3 
.392 199 
•392 695 
•393 191 
•393 685 
•394 179 
•394 673 
•395 l66 
•395 6 5 8 
.396 150 
.396 641 

•397 132 
.397621 

.398 m 
.398 600 
•399 °88 
•399 575 
.400 062 
.400 549 
.401 035 
.401 520 
.402 005 
.402 489 
.402 972 

403 455 
•403 938 
.404 420 
.404 901 
.405 382 
.405 862 
.406 341 
.406 820 
.407 299 
.407 77J 
.408 254 
.408 731 
.409 207 
.409 682 
.410 157 
.410 632 
.411 106 

.411579 
.412052 
.412 524 
.412 996 



D. 1' 



8.45 
8.42 
8.42 
8.42 
8.40 

8.38 
8.38 
8-37 
8-35 
8-35 
8-33 
S-33 
8.32 
8.30 
8.30 
8.28 
8.27 
8.27 
8.27 
8.23 
8.23 
8.23 
8.22 
8.20 
8.20 
8.18 
8.18 
8.15 
8.17 
8.15 

8.13 

8.12 
8.12 
8.12 
8.10 
8.08 
8.08 
8.07 
8.05 
8.05 
8.05 
8.03 
8.02 
8.02 
8.00 
7.98 
7.98 
7.98 
7-97 
7-95 
7-95 
7-93 
7.92 
7.92 
7.92 
7.90 
7.88 
7.88 
7.87 
7.87 

D. 1". 



Cos. 



9.986 904 
.986 873 
.986 841 
.986 809 
.986 778 

9.986 746 
.986 714 
.986 683 
.986651 
.986 619 

9.986587 
•986555 
•986 523 
.986491 
.986 459 

9.986427 
•986 395 
.986 363 
.986 331 
.986 299 

9.986 266 
.986 234 
.986 202 
.986 169 
.986 137 

9.986 104 
.986 072 
.986 039 
.986 007 
•985 974 

9.985 942 

•985 909 
.985 876 
.985 843 
.985811 
9.985 778 

•985 745 
.985 712 
.985 679 
.985 646 
9.985613 
•985 580 
•985 547 
•985 5*4 
.985 480 

9-985 447 
.985 414 

•985 381 
.985 347 
.985 314 

9.985 280 
.985 247 

.985213 
.985 180 
.985 146 
9.985 113 
.985 079 

.985 045 

.985011 

.984 978 

9.984 944 

Sin. 



D. 1". 



•5 2 
•53 
•53 
•5 2 
•53 
•53 
•5 2 
•53 
•53 
•53 
•53 
•53 
•53 
•53 
•53 
•53 
•53 
•53 
•53 
•55 
•53 
•53 
•55 
•53 
•55 
•53 
•55 
•53 
•55 
•53 
•55 
•55 
•55 
•S3 
•55 
•55 
•55 
•55 
•55 
•55 
•55 
•55 
•55 
•57 
•55 
•55 
•55 
•57 
•55 
•57 
•55 
•57 
•55 
•57 
•55 
•57 
•57 
•57 
•55 
•57 

D. 1", 

75^ 



Tan. 



9.396 771 
■397 309 
•397 846 
.398 383 
.398919 

9-399 455 
•399 990 
.400 524 
.401 058 
.401 591 

9.402 124 
.402 656 
.403 187 
.403 718 
.404 249 

9.404 778 
.405 308 
.405 836 
.406 364 
.406 892 

9.407419 

•407 945 
.408471 
.408 996 
.409 521 

9.410 045 
.410 569 
.411 092 
.411 615 
.412137 

9.412 658 

.413 179 
.413699 
.414219 
.414 738 

9415257- 
•415 775 
.416293 
.416 810 
.417326 

9.417842 
.418358 
.418873 
.419387 
.419 901 

9.420415 
.420 927 
.421 440 
.421 952 
.422 463 

9.422 974 
•423 484 
•423 993 
•424 5°3 
.425011 

9425 5 J 9 
.426 027 
426 534 
.427 041 
•427 547 

9.428052 

Cot. 



D. 1". 



8-97 
8.95 
8-95 
8-93 
8-93 
8.92 
8.90 
8.90 



8.87 
8.85 
8.85 
8.85 
8.82 
8.83 
8.80 
8.80 
8.80 
8.78 
8.77 
8.77 
8.75 
8.75 

8-73 

8.73 
8.72 
8.72 
8.70 
8.68 
8.68 
8.67 
8.67 
8.65 
8.65 
8.63 
8.63 
8.62 
8.60 
8.60 
8.60 
8.58 
8-57 
|-S7 
8.57 

8-53 

8.55 

8-53 
8.52 
8.52 
8.50 
8.48 
8.50 
8.47 
8.47 
8.47 
8.45 
8.45 
8-43 
8.42 

D. 1". 



Cot. 



0.603 229 
.602 691 
.602 154 
.601 617 
.601 081 

0.600 545 
.600010 

•599 47° 
.598 942 

•598 409 

0.597 876 

•597 344 
.596813 
.596 282 
•595 751 

0.595 222 
•594 692 
.594 164 
•593 636 
•593 108 

0.592 581 
•592055 

•59i 5 2 9 
.591 004 

•590 479 

o-589 955 

.589431 

.588 908 

.588 385 
.587863 

0.587 342 
.586821 
.586 301 
.585 781 
.585 262 

0.584 743 
.584 225 

•583 707 
.583 190 
•582674 
0.582 158 
.581 642 
.581 127 
•580613 
.580 099 

0-579 585 
.579 073 
.578560 
.578048 
•577 537 

0.577026 

•57° 5 l6 
.576007 

•575 497 

.574 989 

0.574481 

•573 973 
.573466 

•572 959 

•572 453 

Q-57 1 948 

Tan. 



LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 195 

15° 



50 
51 
52 
53 

54 

55 
56 
57 
58 
59 
60 



Sin. 



9.412996 
.413467 

.414408 
.414 878 

9415 347 
•415 8l 5 
.416 283 
.416751 
.417217 

9.417684 
.418 150 
.418 615 
.419079 
•419 544 

9.420 007 
.420 470 
•420 933 
.421 395 
.421 857 

9.422318 
.422 778 
.423 238 
.423 697 
.424156 

9.424615 
•425 o73 
425 53o 
.425 987 
.426 443 

9.426 899 

•427 354 
.427 809 
.428 263 
.428 717 
9.429170 
.429 623 
•430 075 
•43o 527 
.430 978 

9.431 429 
.431 879 

•432 329 
.432 778 
•433 226 

9-433 675 
.434122 

•434 5 6 9 

.435016 

•435 462 

9.435 908 

•436 353 
.436 798 
•437 242 
•437 686 

9.438 129 
.438572 
.439014 
•439 45 6 
•439 897 

9.440338 

Cos. 



D. 1". 



7.85 
7.85 
7-83 
7.83 
7.82 

7.80 
7.80 
7.80 

7-77 
7.78 

7-77 

7-75 
7-73 
7-75 
7.72 

7.72 

7.72 
7.70 
7.70 
7.68 
7.67 
7.67 
7-65 
7-65 
7-65 

7-63 
7.62 
7.62 
7.60 
7.60 
7.58 
7.58 
7-57 
7-57 
7-55 
7-55 
7-53 
7-53 
7-52 
7-5 2 
7-5o 

7-5° 
7.48 

7-47 
7.48 

745 
7-45 
7-45 
7-43 
7-43 
7.42 
7.42 
7.40 
7.40 
7-38 
7-38 
7-37 
7-37 
7-35 
7-35 

D. 1". 



Cos. 



9.984 944 
.984910 
.984 876 
.984 842 
.984 808 

9.984 774 
•984 740 
.984 706 
.984 672 
.984 638 

9.984 603 
.984 569 

.984 535 
.984 500 
.984 466 

9.984 432 
•984 397 
•984 3 6 3 
.984 328 
.984 294 

9.984 259 
.984 224 
.984 190 

•984155 
.984 120 

9.984085 
.984050 
.984015 
.983981 
.983 946 

9.983 91 1 

•983 875 
.983 840 
•983 805 
.983 770 

9-983 735 
.983 700 
.983 664 
.983 629 
.983 594 

9-983 558 
•983 523 
.983 487 
•983452 
.983416 

9.983 381 
•983 345 
•983 309 
•983 273 
.983 238 

9.983 202 
.983 166 
.983 130 
.983 094 
.983058 

9.983 022 
.982 986 
.982 950 
.982914 
.982 878 

9.982 842 
Sin. 



D. 1". 



D.l' 

74° 



Tan. 



9.428 052 
.428 558 
.429 062 
.429 566 
.430 070 

9-43o 573 
43i 075 
43i 577 
.432079 
.432 580 

9.433 080 

433 580 
.434 080 

434 579 

435 078 

9435 57 6 

436 073 
.436 570 

437 o6 7 

437 5 6 3 
9438059 

438 554 

439 048 
439 543 
44° 036 

9.440 529 
.441 022 
.441 514 
.442 006 

442 497 
9.442 988 

443 479 

443 968 
.444458 

444 947 
9445 435 

445 923 
.446411 
.446 898 

447 384 
9.447 870 

448 35 6 
.448 841 

449 326 
.449 810 

9.450 294 

45° 777 
.451 260 

45 1 743 

452 225 
9.452 706 

453 187 

453 668 

454 148 

454 628 

9455 io 7 

455 586 
.456 064 

456 542 
.457019 

9457 496 

Cot. 



D. 1' 



843 
8.40 
8.40 
8.40 
8.38 

8-37 
^37 
8-37 
8-35 
8.33 
8-33 
S-33 
8.32 
8.32 
*8. 3 o 

8.28 
8.28 
8.28 
8.27 
8.27 
8.25 
8.23 
8.25 
8.22 
8.22 
8.22 
8.20 
8.20 
8.18 
8.18 
8.18 
8.15 
8.17 
8.15 
8.13 
8.13 
8.13 
8.12 
8.10 
8.10 
8.10 
8.08 
8.08 
8.07 
8.07 
8.05 
8.05 
8.05 
8.03 
8.02 
8.02 
8.02 
8.00 
8.00 
7.98 
7.98 
7-97 
7-97 
7-95 
7-95 

D. 1". 



Cot. 



0.571 948 
.571442 
.570938 

•570 434 
.569 930 

0.569427 
.568925 
.568 423 
.567921 
.567 420 

0.566 920 
.566420 
.565 920 
.565 421 
•564922 

0.564424 
•563 927 
•563 430 
•562 933 
•562 437 

0.561 941 
.561 446 
.560952 
.560 457 
•559 964 

559 471 
558 978 
558486 

557 994 
557 503 
557oi2 

556521 
556032 

555 542 
555 053 
554 565 
554 077 
553 589 
553 102 
552 616 

552 130 

55 1 644 
55i 159 
55° 674 
55° 190 
549 7°6 
549 223 
548 740 
548 257 
547 775 

547 294 
546813 
546 332 
545 852 
545 372 

544 893 
544414 

543 936 
543 458 
542981 

Q-542 5°4 

Tan. 



60 
59 
58 
57 
56 

55 
54 
53 
52 
5i 
5o 
49 
48 
47 
46 

45 
44 
43 
42 

4i 
40 

39 
38 
37 
36 

35 
34 
33 
32 
3i 
30 
29 
28 
27 
26 

25 

24 

23 
22 
21 



19 
18 

17 
16 

15 
14 
13 
12 
11 
10 

9 
8 



196 LOGARITHMIC SINES, COSINES, TANGENTS, 

16° 



AND COTANGENTS. 



Sin. 



D. 1". 



Cos, 



D. 1", 



Tan, 



D. 1". 



Cot. 



o 

i 

2 

3 
4 

5 
6 

7 
8 

9 

10 

ii 

12 

13 
14 

15 

16 

17 
18 

19 

20 

21 
22 
23 

24 

25 
26 

27 
28 

29 

30 

3i 
32 
33 
34 

35 
36 
37 
38 
39 
40 
4i 
42 
43 
44 

45 
46 

47 
48 

49 
5o 
5i 
52 
53 
54 
55 
56 
57 
58 
59 
60 



9.440 338 
.440 778 
.441 218 
.441 658 
.442 096 

9-442 535 

442 973 
•443 4io 
•443 847 
.444 284 

9.444 720 
•445 155 
•445 590 
.446 025 

•446 459 
9.446 893 
•447 326 
•447 759 
.448 191 
.448 623 
9.449 054 
•449 485 
•449 915 
•45° 345 
•450 775 
9.451 204 
.451 632 
.452 060 
.452488 
.452915 

9-453 342 
•453 768 
•454 194 
.454619 

•455 °44 

9455 469 

•455 893 

.456316 

45 6 739 

457 l62 

9457 5 8 4 
.458 006 
.458 427 

458 848 
.459 268 

9459 688 
.460 108 
.460 527 
.460 946 
.461 364 

9.461 782 
.462 199 
.462 616 
.463 032 
.463 448 

9.463 864 
.464 279 
.464 694 
.465 108 
.465 522 

9465 935 



7-33 
7-33 
7-33 
7-3° 
7-32 

7-3o 

7.28 
7.28 
7.28 

7.27 

7-25 
7-25 
7-25 
7-23 
7-23 
7.22 
7.22 
7.20 
7.20 
7.18 
7.18 
7.17 
7.17 
7.17 
7-i5 
7-i3 
7*3 
7i3 
7.12 
7.12 
7.10 
7.10 
7.08 
7.08 
7.08 
7.07 
7-o5 
7-05 
7-o5 
7-°3 

7-°3 
7.02 
7.02 
7.00 
7.00 
7.00 
6.98 
6.98 
6.97 
6.97 

6-95 
6-95 
6-93 
6-93 
6-93 
6.92 
6.92 
6.90 
6.90 
6.88 

D. 1". 



982 842 
982 805 
982 769 
982 733 
982 696 

982 660 
982 624 
982587 
982551 
982514 
982477 
982 441 
982 404 
982 367 
982331 
982 294 
982 257 
982 220 
982 183 
982 146 
982 109 
982 072 
982035 
981 998 
981 961 
981 924 
981 886 
981 849 
981 812 
981 774 

981 737 
981 700 
981 662 
981 625 
981 587 
981 549 
981 512 
981 474 
981 436 
981 399 
981 361 
981 323 
981 285 
981 247 
981 209 
981 171 
981 133 
981 095 
981057 
981 019 
980 981 
980 942 
980 904 
980 866 
980 827 
980 789 
980 750 
980 712 
980 673 
980 635 

980 596 
Sin. 



.62 
.60 
.60 
.62 
.60 
.60 
.62 
.60 
.62 
.62 
.60 
.62 
.62 
.60 
.62 
.62 
.62 
.62 
.62 
.62 
.62 
.62 
.62 
.62 
.62 

•63 

.62 
.62 

.63 
.62 

.62 

.63 
.62 

•63 
.63 
.62 
.63 
.63 
.62 

.63 
.63 
.63 
.63 
.63 
.63 
.63 
•63 
.63 
.63 
•63 

i s 
•63 

.63 
.65 

•63 
.65 

•63 
.65 

.63 
.65 

D. 1". 
73° 



9457 496 

457 973 

458 449 

458925 

459 400 

9459 875 
.460 349 
.460 823 
.461.297 
.461 770 

9.462 242 
.462 715 
.463 186 
.463 658 
.464 128 

9.464 599 
.465 069 

465 539 
.466 008 
.466477 

9.466 945 
.467413 
.467 880 
.468 347 
.468814 

9.469 280 
.469 746 
.470 211 
.470 676 
.471 141 

9.471 605 
.472 069 
.472532 

472 995 

473 457 
9473 919 

474 381 

474 842 

475 303 
475 763 

9.476 223 
.476 683 

477 142 

.477 601 
.478 059 

9478 517 

478 975 

479 432 
.479 889 
.480 345 

9.480 801 
.481 257 
.481 712 
.482 167 
.482 621 

9483 075 
483 529 

483 982 

484 435 
.484 887 

9485 339 

Cot. 



7-95 
7-93 
7-93 
7.92 
7.92 
7.90 
7.90 
7.90 
7.88 
7.87 
7.88 
7.85 
7.87 

7-83 
7.85 

7-83 
7-83 
7.82 
7.82 
7.80 
7.80 
7.78 
7.78 
7.78 
7-77 
7-77 
7-75 
7-75 
7-75 
7-73 

7-73 
7.72 
7.72 
7.70 
7.70 
7.70 
7.68 
7.68 
7.67 
7.67 
7.67 
7-65 
7-65 
7-63 
7-63 

7-63 

7.62 
7.62 
7.60 
7.60 
7.60 
7.58 
7.58 
7-57 
7-57 
7-57 
7-55 
7-55 
7-53 
7-53 

D. 1". 



o-542 504 
•542027 

•541 55 1 
•54i 075 
.540 600 

c.540 125 
•539651 
•539 177 
•538 703 
.538 230 

°-537 758 
•537 285 
.536814 
.536 342 
■535 872 

o.535 4oi 
•534 931 
•534 46i 
•533 992 
•533 5 2 3 
533055 

532587 
532 120 
531653 
53i 186 
53o 720 
530254 
529 789 
529 324 
528859 
528 395 
527 931 
527468 
527 005 
526543 
526081 
525619 
525 158 
524697 

524 237 

523 777 
523317 
522858 
522 399 
521 941 
521 483 
521 025 
520 568 
520 in 
5^655 
5 J 9i99 
5^743 
518288 

5^833 
5*7 379 

5 l6 925 
516471 
516018 
5^565 
5i5 "3 
514 661 

Tan. 



LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 197 

17° 



M, 



D. 1' 



o 


9-465 935 


I 


.466 348 


2 


.466 761 


3 


.467 173 


4 


•467 585 


5 


9.467 996 


6 


.468 407 


7 


.468817 


8 


.469 227 


9 


.469 637 


IO 


9.470 046 


ii 


•47°455 


12 


.470 863 


13 


.471 271 


14 


•47 1 679 


15 


9.472086 


16 


.472 492 


17 


.472 898 


18 


•473 304 


19 


•473 7 10 


20 


9-474 "5 


21 


•474 5 *9 


22 


•474 923 


23 


•475 327 


24 


•475 73o 


25 


9-47 6 *33 


26 


.476 536 


27 


.476 938 


28 


•477 340 


29 


•477 741 


30 


9.478 142 


31 


.478 542 


32 


.478 942 


33 


•479 342 


34 


•479 74i 


35 


9.480 140 


36 


•480 539 


37 


.480 937 


38 


•481 334 


39 


.481 731 


40 


9.482 128 


4i 


482 525 


42 


.482 921 


43 


.483316 


44 


483 7 12 


45 


9.484 107 


46 


.484 501 


47 


.484 895 


48 


.485 289 


49 


.485 682 


50 


9.486 075 


5i 


.486 467 


52 


.486 860 


53 


.487251 


54 


.487 643 


55 


9.488 034 


56 


.488 424 


57 


.488814 


58 


.489 204 


59 


489 593 


60 


9.489 982 



Cos. 



6.88 
6.88 
6.87 
6.87 
6.85 
6.85 
6.83 
6.83 
6.83 
6.82 
6.82 
6.80 
6.80 
6.80 
6.78 
6.77 
6.77 
6.77 
6.77 
6-75 
6-73 
6-73 
6-73 
6.72 
6.72 
6.72 
6.70 
6.70 
6.68 
6.68 
6.67 
6.67 
6.67 
6.65 
6.65 
6.65 
6.63 
6.62 
6.62 
6.62 
6.62 
6.60 
6.58 
6.60 
6.58 

6-57 
6.57 

6 -57 
6.55 
6.55 

6-53 
6.55 
6.52 

6-53 
6.52 

6.50 
6.50 
6.50 

6.48 
6.48 

D, 1". 



Cos. 



9.980 596 
•980558 
.980 519 
.980 480 
.980 442 

9.980 403 
.980 364 
•980 325 
.980 286 
.980 247 

9.980 208 
.980 169 
.980 130 
.980091 
.980052 

9.980012 
•979 973 
•979 934 
•979 895 
•979 855 

9.979816 
•979 77 6 
•979 737 
•979 697 
.979658 

9.979618 
•979 579 
•979 539 
•979 499 
•979 459 

9.979 420 
•979 380 
•979 340 
•979 3°° 
.979 260 

9.979 220 
.979 180 

•979 HO 
.979 100 

•979°59 

9.979019 

.978 979 

•978 939 
.978 898 
.978858 

9.978817 
•978 777 
•978 737 
.978 696 
•978 655 

9.978615 
•978 574 
•978 S33 
•978 493 
.978452 

9.978 41 1 

■978 37° 

.978 329 

.978 288 

.978 247 

9.978 206 

Sin! 



D. V 



•63 
.65 
.65 

.63 
.65 

.65 
.65 
.65 
.65 
.65 
.65 
.65 
.65 
.65 
.67 
.65 

•65 
.65 
.67 
•65 
.67 
.65 
.67 
.65 
•67 
.65 
.67 
.67 
.67 
.65 
.67 
.67 
.67 
.67 
.67 
.67 
.67 
.67 
.68 
.67 
.67 
.67 
.68 
.67 
.68 

.67 
.67 
.68 
.68 
.67 
.68 
.68 
.67 
.68 
.68 
.68 
.68 
.68 
.68 
.68 

D. 1" 

72° 



Tan. 



9485 339 
.485 791 
.486 242 
.486 693 
.487 143 

9487 593 
.488 043 
.488 492 
.488 941 

489 39o 
9.489 838 

.490 286 

490 733 
.491 180 
.491 627 

9.492 073 
.492519 
.492 965 
.493 410 

493 854 
9.494 299 

494 743 
.495 186 

495 6 3° 
.496073 

94965*5 

496 957 

497 399 
497 841 
.498 282 

9.498 722 
499 163 
499 603 
.500 042 
.500481 

9.500 920 

•5 DI 359 
.501 797 

•5°2 235 
.502 672 

9.503 109 
.503 546 
•5°3 982 
.504418 

•5°4 854 
9.505 289 

•5°5 724 
.506159 

•5°6 593 
.507 027 

9.507 460 

•5°7 893 
.508 326 

•5°8 759 

•5°9 191 

9.509 622 

•5 IO °54 
.510485 
.510 916 
.511 346 

9-5 11 77 6 
Cot. 



D. 1' 



7-53 
7-5 2 
7-52 
7-5o 
7-5° 

7-5o 
7.48 
7.48 
7.48 
747 
747 
745 
745 
745 
743 
743 
743 
7.42 
7.40 
7.42 
7.40 
7-3^ 
7.40 
7-38 
7-37 
7-37 
7-37 
7-37 
7-35 
7-33 
7-35 
7-33 
7-32 
7-32 
7-32 
7-32 
7-3o 
7-3o 
7.28 
7.28 
7.28 
7.27 
7.27 
7.27 
7-25 
7-25 
7-25 
7-23 
7-23 
7.22 

7.22 
7.22 
7.22 
7.20 
7.18 
7.20 
7.18 
7.18 
7.17 
7.17 

D. 1". 



Cot. 



0.514 661 
.514209 
•513 758 
•5 I 3 3°7 
•5*2857 

0.512407 

•5 11 957 

.511 508 

•5 iio 59 
.510 610 

0.510 162 
.509 714 
.509 267 
.508 820 
•5°8 373 

0.507 927 
.507481 

•507035 

.506 590 

.506 146 

0.505 701 

•5°5 257 
.504814 

•5°4 37° 
•503927 

0.503 485 
•5°3 043 
.502 601 
.502 159 
.501 718 

0.501 278 
.500 837 
.500 397 
.499958 
499 5*9 

0.499 0S0 
.498 641 
.498 203 
497 765 
497 328 

0.496 891 
496 454 
.496018 
495 582 
495 H6 

0.494 711 
494 276 
.493841 
493 407 
492 973 

0.492 540 
.492 107 
.491 674 
.491 241 
.490 809 

0.490 378 
489 946 
.489515 
.489 084 
488 654 

0.488 224 
Tan. 



60 
59 
58 
57 
56 

55 

54 
53 
52 
5i 
5o 
49 
48 
47 
46 

45 
44 
43 
42 
4i 
40 
39 
38 
37 
36 

35 
34 
33 
32 
3i 
30 
29 
28 
27 
26 

25 

24 
23 

22 



1 9 
18 

*7 
16 

15 
14 
13 
12 
11 
10 

9 
8 



o 

"ST 



198 LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 

18° 



Sin. 



o 


9.489 982 


I 


.490 371 


2 


•490 759 


3 


.491 147 


4 


•491 535 


5 


9.491 922 


6 


.492 308 


7 


.492 695 


8 


.493081 


9 


•493 466 


IO 


9.493851 


ii 


•494 236 


12 


.494 621 


13 


•495 °°5 


H 


•495 388 


15 


9-495 772 


16 


.496 154 


17 


496 537 


18 


.496919 


19 


•497 301 


20 


9.497 682 


21 


.498 064 


22 


•498 444 


23 


•498 825 


24 


.499 204 


25 


9.499 584 


26 


•499 963 


27 


.500 342 


28 


.500 721 


29 


.501 099 


30 


9.501 476 


31 


.501 854 


32 


.502 231 


33 


.502 607 


34 


.502 984 


35 


9-5°3 360 


3b 


•5°3 735 


37 


.504110 


38 


•5°4 485 


39 


.504 860 


40 


9-5°5 234 


4i 


.505 608 


42 


.505 981 


43 


•506 354 


44 


.506 727 


45 


9.507 099 


4 b 


.507471 


47 


•5°7 843 


48 


.508 214 


49 


.508585 


50 


9.508 956 


5i 


.509 326 


52 


.509 696 


53 


.510065 


54 


•5 I0 434 


55 


9.510803 


5& 


.gn 172 


57 


•5" 540 


58 


.511907 


59 


.512275 


60 


9.512 642 



D. 1". 



Cos. 



6.48 
6.47 
6.47 
6.47 
6-45 
6-43 
6-45 
6-43 
6.42 
6.42 
6.42 
6.42 
6.40 
6.38 
6.40 

6-37 
6.38 

6-37 
6-37 
6-35 
6-37 
6-33 
6.35 
6.32 

6.33 
6.32 
6.32 
6.32 
6.30 
6.28 
6.30 
6.28 
6.27 
6.28 
6.27 
6.25 
6.25 
6.25 
6.25 
6.23 
6.23 
6.22 
6.22 
6.22 
6.20 
6.20 
6.20 
6.18 
6.18 
6.18 
6.17 
6.17 
6.15 
6.15 
6.15 

6.15 
6.13 
6.12 

6.13 
6.12 

D. 1". 



Cos. 



9.978 206 
.978 165 
.978 124 
.978083 
.978 042 

9.978 001 

•977 959 
.977918 
.977877 
•977 835 
9-977 794 
•977 752 
.977711 
•977 669 
.977 628 

9.977 586 
•977 544 
•977 5°3 
.977461 
.977419 

9-977 377 
•977 335 
•977 293 
.977251 

•977 209 
9.977 167 
•977 125 
.977083 
.977 041 
.976999 

9-97 6 957 
.976914 
.976872 
.976 830 
.976 787 

9-976 745 
.976 702 
.976 660 
.976 617 
.976 574 

9-976 532 
.976489 
.976 446 
•976 404 
.976361 

9.976318 
•976 275 
.976 232 
.976 189 
.976 146 

9.976 103 
.976 060 
.976017 
•975 974 
•975 930 

9.975 887 
•975 844 
•975 800 
•975 757 
•975 7H 

9.975 670 



D. 1' 



.68 

.68 

.68 

.68 

.68 

.70 

.68 

.68 

.70 

.68 

.70 

.68 

.70 

.68 

.70 

.70 

.68 

.70 

.70 

.70 

.70 

.70 

.70 

.70 

.70 

.70 

.70 

.70 

.70 

.70 

.72 

.70 

■70 

.72 

.70 

.72 

.70 

.72 

.72 

.70 

.72 

.72 

.70 

.72 

.72 

.72 

.72 

.72 

.72 

•72 

.72 

•72 
.72 

•73 

.72 

.72 

•73 
.72 
.72 
•73 

7P" 



Tan. 



9.511776 
.512 206 
•5 1 2 635 
.513064 
•5 1 3 493 

9-5*3 921 
•5H349 
•5H777 
.515204 

•5J5 63I 

9.516057 

.516484 

.516910 

•517 335 
.517761 

9.518 186 
.518610 

•5 1 9 034 
.519458 
.519882 

9.520 305 
.520 728 
.521 151 
•521 573 
•521 995 

9.522417 
.522838 

•523259 
.523 680 
.524 100 

9.524520 
.524940 
•5 2 5 359 
•525 778 
•526 197 

9.526 615 
•5 2 7 033 
•5 2 7 45 1 
.527 868 
.528 285 

9.528 702 
.529119 
•529 535 
•529 95 1 
.530 366 

9-530 78I 
•53i 196 

.531611 
.532025 
•532 439 

9-532853 
•533 266 
•533 679 
•534 092 
•534 5°4 

9-534 9i6 
•535 328 
•535 739 
•536150 
•536561 

9-536 972 
Cot. 



D. 1". 



7.17 
7-15 
7-15 
7-15 
7-i3 
7-i3 
7-13 
7.12 
7.12 
7.10 
7.12 
7.10 
7.08 
7.10 
7.08 
7.07 
7.07 
7.07 
7.07 
7-05 
7-o5 
7-05 
703 
7-03 
7-°3 
7.02 
7.02 
7.02 
7.00 
7.00 
7.00 
6.98 
6.98 
6.98 
6.97 

6-97 
6.97 

6-95 

6-95 

6-95 

6-95 

6-93 

6-93 

6.92 

6.92 

6.92 

6.92 

6.90 

6.90 

6.90 

6.88 

6.88 

6.88 

6.87 

6.87 

6.87 

6.85 

6.85 

6.85 

6.85 

D. 1". 



Cot. 




0.488 224 


60 


.487 794 


59 


.487 365 


58 


.486 936 


57 


.486 507 


5b 


0.486 079 


55 


•485 651 


54 


.485 223 


53 


.484 796 


52 


.484 369 


5i 


0483 943 


50 


.483516 


49 


.483 090 


48 


.482 665 


47 


.482 239 


4b 


0.481 814 


45 


•481 390 


44 


.480 966 


43 


.480 542 


42 


.480118 


4i 


0.479 695 


40 


•479 272 


39 


•478 849 


38 


478 427 


37 


.478 005 


3b 


0-477 583 


35 


.477 162 


34 


.476 74i 


33 


.476320 


32 


•475 900 


3i 


o.475 48o 


30 


.475 060 


29 


•474 641 


28 


.474 222 


27 


•473 803 


26 


0-473 385 


25 


•472 967 


24 


•472 549 


23 


.472 132 


22 


•471 715 


21 


0.471 298 


20 


.470 881 


19 


.470 465 


18 


.470 049 


17 


•469 634 


16 


0.469 219 


15 


.468 804 


14 


.468 389 


13 


•467 975 


12 


.467 561 


11 


0.467 147 


10 


•466 734 


9 


.466 321 


8 


.465 908 


7 


•465 496 


6 


0.465 084 


5 


.464 672 


4 


.464 261 


3 


•463 850 


2 


•463 439 


1 


0.463 028 





Tan. 


M. 



LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 199 

19° 



M, 



o 
i 

2 

3 
4 

5 
6 

7 
8 

9 
io 
ii 

12 

13 
14 

15 
16 

17 
18 

19 

20 

21 
22 
23 

24 

25 
26 

27 
28 

29 
30 
3i 

32 

33 
34 

35 
36 
37 
38 
39 
40 

4i 
42 

43 
44 

45 
46 

47 
48 

49 
50 
5i 
52 
53 
54 

55 
56 
57 
58 
59 
60 



9- 



512 642 
513009 

513 375 
5*3 741 
5*4 107 
5*4472 
5H837 
515 202 
515 566 
5i5 93o 
516294 
516657 
5:7 020 
5*7382 
5*7 745 
5*8 107 
518468 
518829 
5*9190 
5*955* 

5*99*1 
520 271 
520631 

520 990 

521 349 

521 707 

522 066 
522424 

522 781 

523 138 

523 495 
523852 

524 208 
,524 564 
,524920 

525 275 
525 630 

525 984 

526 339 
,526693 

527046 
,527 400 

■527 753 
,528 105 
,528458 
,528810 
•5 2 9 161 

•529 513 
.529 864 
•530215 

.530 565 
■530 915 
•53* 265 
•53*614 
•53* 963 
.532312 
.532661 
•533009 
•533 357 
.533704 
•534 052 
Cos. 



D. 1". 



6.12 

6.10 
6.10 
6.10 
6.08 
6.08 
6.08 
6.07 
6.07 
6.07 
6.05 
6.05 
6.03 
6.05 
6.03 
6.02 
6.02 
6.02 
6.02 
6.00 
6.00 
6.00 
5.98 
5.98 
5-97 
5.98 
5-97 
5-95 
5-95 
5-95 
5-95 
5-93 
5-93 
5-93 
5-92 

5-92 
5-90 
5-92 
5-9o 
5.88 

5-9o 
5.88 
5.87 
5.88 
5.87 
5.85 
5.87 
5.85 
5.85 
5-83 
5-83 
5.83 
5.82 
5.82 
5.82 
5.82 
5.80 
5.80 
5.78 
5.80 

D. 1" 



Cos. 



9.975 670 
•975 627 
•975 583 
•975 539 
•975 496 

9-975 452 
•975 408 
•975 365 
•975 321 
•975 277 

9-975 233 
•975 * 8 9 
•975 *45 
•975 io* 
•975 057 

9-975 013 
•974 969 
•974 925 
.974 880 
.974836 

9.974 792 
•974 748 
•974 7°3 
•974 659 
.974614 

9-974 57° 
•974 525 
.974 481 
.974436 

•974 39* 
9-974 347 
•974 302 
•974 257 
.974212 

•974 167 
9.974 122 

•974 077 
.974032 

•973 987 
•973 942 

9-973 897 
•973 852 
.973 807 

•973 761 
.973716 

9.973671 
•973 625 
•973 580 
•973 535 
•973 489 

9-973 444 
•973 398 
•973 352 
•973 307 
.973 261 

9-973 2I5 

•973 169 
.973 124 
.973078 
•973 032 
9.972 986 



D. 1". 



.72 
•73 
•73 

.72 

•73 

•73 

.72 

•73 
•73 

•73 
•73 
•73 
•73 
•73 
•73 
•73 
•73 
•75 
•73 
•73 
•73 
•75 
•73 
•75 
•73 

•75 
•73 
•75 
•75 
■73 

•75 
•75 
•75 
•75 
•75 
•75 
•75 
•75 
•75 
•75 
•75 
•75 
•77 
•75 
•75 
•77 
•75 
•75 
•77 
•75 
•77 
•77 
•75 
•77 
•77 
•77 
•75 
•77 
•77 
•77 

dTI 7 - 
70° 



Tan. 



9.536 972 
•537 382 
•537 792 
.538 202 
.538611 

9.539020 
•539 429 
•539 837 
•540 245 
•540 653 

9.541 061 

. .541 468 

•54i 875 
.542 281 
.542 688 

9-543 094 
•543 499 
•543 905 
•544 3*0 
•544 7*5 

9-545 **9 
•545 524 
•545 928 
•546 331 
•546 735 

9-547 138 
•547 540 
•547 943 
•548 345 
•548 747 

9-549 *49 
•549 55° 
•549 95* 
•550 352 
•550 752 

9-55* *53 

•55*552 
•55*952 
•552351 
•552 750 

9-553 *49 

•553 548 
•553 946 
•554 344 
•554 74* 

9-555 *39 
•555 536 
•555 933 
.556 329 
•556725 

9.557 121 
•557 5*7 
•557 9*3 
.558308 

•558703 

9.559 097 
•559 49* 
.559885 
.560 279 
.560 673 

9.561 066 
Cot. 



D. 1". 



6.83 
6.83 
6.83 
6.82 
6.82 
6.82 
6.80 
6.80 
6.80 
6.80 
6.78 
6.78 
6.77 
6.78 
6-77 
6-75 
6-77 
6-75 
6-75 
6-73 
6.75 
6-73 
6.72 

6-73 
6.72 

6.70 
6.72 
6.70 
6.70 
6.70 
6.68 
6.68 
6.68 
6.67 
6.68 
6.65 
6.67 
6.65 
6.65 
6.65 
6.65 
6.63 
6.63 
6.62 
6.63 
6.62 
6.62 
6.60 
6.60 
6.60 
6.60 
6.60 
6.58 
6.58 
6.57 

6-57 

6.57 
6.57 
6-57 
6-55 

D. 1". 



Cot. 



0.463 028 
.462618 
.462 20S 
.461 798 
.461 389 

0.460 980 
.460571 
.460 163 
•459 755 
•459 347 

0.458 939 
•458 532 
•458 125 
•457 7*9 
•457 3*2 

0.456 906 
•456 5°* 
•456095 
•455 690 
•455 285 

0.454881 
.454 476 
.454072 
•453 669 
•453 265 

0.452 862 
.452 460 
•452057 
•45*655 
•45* 253 

0.450 851 
•45° 45° 
•45° °49 
.449 648 

•449 248 

0.448 847 

.448 448 

.448 048 

•447 649 

•447 250 

0.446851 

•446452 

.446 054 

•445 656 

•445 259 

0.444 861 

•444 464 
.444 067 
.443671 
•443 275 

0.442 879 
.442 483 
.442 087 
.441 692 
•44* 297 

0.440 903 
.440 5°9 
.440115 

•439 72* 

•439 327 

o-438 934 

Tan. 



60 
59 
58 
57 
56 

55 

54 
53 
52 
5i 
50 
49 
48 
47 
46 

45 
44 
43 
42 

4i 
40 
39 
38 
37 
36 
35 
34 
33 
32 
3i 
30 
29 
28 
27 
26 

25 
24 

23 
22 
21 
20 

19 
18 

17 
16 

15 
*4 
13 
12 
11 
10 

9 
8 

7 
6 

5 
4 
3 
2 



200 LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 

20° 



M, 



Sin. 



o 


9-534 052 . 


I 


•534 399 


2 


•534 745 


3 


•535 °92 


4 


•535 438 


5 


9-535 783 


6 


.536129 


7 


•536 474 


8 


.536818 


9 


•537 163 


10 


9-537 507 


ii 


•537 851 


12 


•538 194 


13 


•538 538 


14 


.538 880 


15 


9-539 223 


16 


•539 565 


17 


•539 907 


18 


.540 249 


19 


•54o 590 


20 


9-540 93I 


21 


.541 272 


22 


•54i 613 


23 


•54i 953 


24 


.542 293 


25 


9.542 632 


26 


•542 97i 


27 


•543 3io 


28 


•543 649 


29 


•543 987 


30 


9-544 325 


31 


•544 663 


32 


.545 000 


33 


•545 338 


34 


•545 6 74 


35 


9.546 01 1 


36 


•546 347 


37 


.546683 


38 


.547019 


39 


•547 354 


40 


9.547 689 


4i 


.548 024 


42 


•548 359 


43 


•548 693 


44 


•549027 


45 


9.549 360 


46 


•549 693 


47 


.550026 


48 


•550 359 


49 


.550692 


50 


9-55 IQ2 4 


5i 


•551356 


52 


.551687 


53 


.552018 


54 


•552 349 


55 


9.552680 


56 


•553 0io 


57 


•553 341 


58 


•553 670 


59 


.554000 


60 


9-554 329 



D. 1' 



5.78 


5-77 


5-78 


5-77 


5-75 


5-77 


5-75 


5-73 


5-75 


5-73 


5-73 


5-7 2 


5-73 


5-7° 


5-72 


5-7o 


5-7o 


5-7o 


5.68 


5.68 


5.68 


5.68 


5- 6 7 


5- 6 7 


5.65 


5.65 


5.65 


5.65 


5-63 


5-63 


5.63 


5.62 


5-63 


5.60 


5.62 


5.60 


5.60 


5.60 


5-58 


5-58 


5.58 


5.58 


5-57 


5-57 


5-55 


5-55 


5-55 


5-55 


5-55 


5-53 


5-53 


5-52 


5-52 


5-52 


5-5 2 


5-5° 


5-52 


5.48 


5-5° 


548 



Cos. 



Cos. 



D. 1". 



9.972 986 
.972 940 
.972 894 
.972 848 
.972 802 

9-972 755 
.972 709 
.972 663 
.972 617 
•972 57° 

9.972524 
.972478 
.972431 
•972 385 
•972 338 

9.972 291 
.972 245 
.972 198 
.972151 
.972 105 

9.972058 
.972 on 

•97 1 964 
.971 917 
.971 870 

9.971 823 
.971 776 
.971 729 
.971 682 
•97 1 6 35 

9.971 588 
.971 540 
•97i 493 
•97 l 446 
.971 398 

9-971 35 1 
.971 303 
.971 256 
.971 208 
.971 161 

9.971 113 
.971 066 
.971 018 
.970970 
.970922 

9.970 874 
.970 827 

•97o 779 
.970731 
.970683 

9-97° 635 
.970 586 
.970 538 
.970 490 
.970442 

9-97° 394 
•97° 345 
.970 297 

•97° 249 
.970 200 

9-97° *5 2 
Sin. 



D. 1". 



77 
77 
77 
77 
78 

77 
77 
77 
78 
77 

77 
78 
77 
78 
78 

77 
78 
78 

77 
78 
78 
78 
78 
7S 
7S 
7S 
7S 
78 
7S 
7S 
So 

/S 
78 
So 
7S 
So 

7S 
So 
7S 
So 

78 
So 
So 
So 
So 

73 
So 
So 
So 
So 
S2 
So 
So 
So 
So 
82 
So 
So 
82 
80 

D. 1", 



69° 



Tan. 



561 066 
5 6l 459 

561 851 

562 244 

562 636 

563 028 

5 6 34i9 

563 811 

564 202 
564 593 

564 983 

565 373 

565 763 
566153 

566 542 

566 932 

567 320 

567 7°9 

568 098 

568 486 

568873 

569 261 

569 648 
570035 
570422 

570 809 

57i 195 

57i 581 

571 967 

572 35 2 

572 738 

573 123 

573 507 

573 892 
574276 

574 660 

575 °44 
575 427 

575 8io 

576 193 
576576 

576 959 

577 341 

577 723 

578 104 
578486 
578867 

579 248 
579629 
580009 

580 389 

580 769 

581 149 
581 528 

581 907 

582 286 

582 665 

583 044 
583422 

583 800 

584 177 
Cot. 



D. 1". 



6-55 
6-53 
6-55 
6.53 
6-53 
6.52 

6-53 
6.52 
6.52 
6.50 
6.50 
6.50 
6.50 
6.48 
6.50 
6.47 
6.48 
6.48 
6-47 
6-45 
6.47 
6.45 
6-45 
6-45 
6-45 
6-43 
6-43 
6-43 
6.42 

6-43 
6.42 
6.40 
6.42 
6.40 
6.40 
6.40 
6.38 
6.38 
6.38 
6.38 
6.38 
6-37 
6-37 
6-35 
6-37 
6-35 
6-35 
6-35 
6-33 
6-33 
6-33 
6-33 
6.32 
6.32 
6.32 
6.32 
6.32 
6.30 
6.30 
6.28 

D. 1". 



Cot. 



0.438 934 
438 541 
.438 149 

•437 756 
•437 364 

0.436 972 
.436 581 
.436 189 
•435 798 
•435 407 

0.435017 
•434 627 
•434 237 
•433 847 
•433 458 

0.433 068 
.432 680 
.432 291 
431 902 
•431 5H 

0.431 127 
•430 739 
•430 352 
•429 965 
•429 578 

0.429 191 
.428 805 
.428419 
.428 033 
.427 648 

0.427 262 
.426 877 
.426 493 
.426 108 
•425 724 

0.425 340 
.424 956 

•424 573 
.424 190 
.423 807 

0.423 424 
.423 041 
.422 659 
.422 277 
.421 896 

0.421 514 

•421 133 
.420 752 
.420371 
.419991 

0.419 611 
.419-231 
.418851 
.418472 
.418093 

0.417 714 

•417 335 
.416 956 
.416578 
.416 200 

0.415 823 

Tan, 



LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 201 

21° 



Sin, 






9-554 329 


I 


•554 658 


2 


.554 987 


3 


•555 315 


4 


•555 643 


5 


9-555 97i 


6 


•55 6 299 


7 


.556 626 


8 


•556 953 


9 


•557 280 


IO 


9.557 606 


ii 


•557 932 


12 


.558258 


13 


•558 583 


14 


•558 909 


.15 


9-559 234 


16 


•559 558 


17 


•559 883 


18 


.560 207 


19 


•5 6o 53i 


20 


9.560 855 


21 


.561 178 


22 


.561 501 


23 


.561 824 


24 


.562 146 


2 5 


9.562468 


26 


.562 790 


27 


.563 112 


23 


•5 6 3 433 


29 


•563 755 


30 


9.564075 


31 


•564 396 


32 


.564716 


33 


•565 036 


34 


•565 356 


35 


9.565 676 


36 


•565 995 


37 


.566314 


38 


.566 632 


39 


•566 95 1 


40 


9.567 269 


4i 


.567587 


42 


•567 904 


43 


.568 222 


44 


•568 539 


45 


9.568 856 


46 


.569 172 


47 


.569 488 


48 


.569 804 


49 


.570 120 


50 


9.570 435 


5i 


•570 75I 


52 


.571 066 


53 


.571 380 


54 


•57i 695 


55 


9.572009 


5«> 


.572 323 


57 


•572636 


58 


•572 950 


59 


•573 263 


60 


9-573 575 



Cos. 



D, 1' 



5-48 
5.48 

5-47 

5-47 
5-47 
5-47 
5-45 
5-45 
5-45 
5-43 
5-43 
5-43 
5-42 
5-43 
5-42 

54o 
542 
540 
5-40 
540 
5-38 
5-38 
5-38 
5-37 
5-37 
5-37 
5-37 
5-35 
5-37 
5-33 
5-35 
5-33 
5-33 
5-33 
5-33 
5-32 
5-32 
5-3° 
5.32 
5.30 

5-3o 

5.28 
5.30 
5.28 
5.28 

5-27 
5-27 
5- 2 7 
5-27 
5- 2 5 
5-27 
5- 2 5 
5-23 
5-25 
5-23 

5- 2 3 
5.22 

5.23 

5.22 
5.20 

D. 1". 



Cos. 



9.970152 
.970 103 

•970 055 
.970 006 

.969 957 
9.969 909 
.969 860 
.969 811 
.969 762 
.969 7H 
9.969 665 
.969 616 

.969 5 6 7 

.969 518 

.969 469 

9.969 420 

•969 37° 
.969321 
.969 272 
.969 223 

9.969173 
.969 124 
.969075 
.969 025 
.968 976 

9.968926 
.968877 
.968 827 
•968 777 
.968 728 

9.968 673 
.968 628 
.968578 
.968 528 
•968 479 

9.968429 
.968 379 
.968 329 
.968 278 
.968 228 

9.968 178 
.968 128 
.968 078 
.968 027 
•967 977 

9.967927 
.967876 
.967 826 

•967 775 
.967 725 

9.967 674 
.967 624 

•967 573 
.967 522 
.967471 

9.967421 
.967 370 
.967319 
.967 268 
.967217 

9.967 166 
Sin. 



D. 1' 



D. 1". 

68°" 



Tan. 



9-584 177 
•584 555 
.584932 

•585 309 
.585 686 

9.586062 

•5 S6 439 
•586815 
.587 190 
.587 566 

9.587941 
.588316 
.588691 
.589066 
.589 440 

9.589814 
.590 188 
.590 562 

.590 935 
.591 308 

9.591 681 
•592054 
.592426 
•592 799 
•593 I7 1 

9-593 542 
•593 9H 
•594 285 
.594656 

•595 027 
9.595 398 
•595 768 
.596138 
.596 508 
.596878 

9-597 247 
.597 616 

•597 985 

•598 354 

•598 722 

9.599091 

•599 459 
.599827 
.600 194 
.600 562 

9.600 929 
.601 296 
.601 663 
.602029 
.602 395 

9.602 761 
.603 127 
•603 493 
.603 858 
.604 223 

9.604 588 

•604 953 
.605317 
.605 682 
.606 046 
9.606410 

Cot. 



D. 1". 



6.28 
6.28 
6.28 
6.27 
6.28 
6.27 
6.25 

6.25 

6.25 
6.25 
6.25 
6.23 
6.23 

6.23 
6.23 

6.22 
6.22 
6.22 
6.22 
6.20 
6.22 
6.20 
6.18 
6.20 
6.18 
6.18 
6.18 
6.18 
6.I7 
6.I7 
6.I7 
6.I7 
6.I5 
6.I5 
6.I5 
6.I5 
6.I3 
6.I5 

6.13 
6.13 
6.12 

6.13 

6.12 
6.12 
6.12 

6.10 
6.10 
6.10 
6.10 
6.10 

6.08 
6.08 
6.08 
6.08 

6.07 

6.0S 

6.07 
6.07 

D. 1". 



Cot. 



0.415823 

415 445 
.415 068 
.414691 
.414314 

0.413 938 
413 56i 
413 185 
.412 810 
.412434 

0.412059 
.411 684 
.411309 
.410934 
.410 560 

0.410 186 
.409 812 

•409 438 
.409 065 
.408 692 

0.408 319 
•407 946 
407 574 
.407 201 
.406 829 

0.406458 
.406 086 

405 7I5 
405 344 
404 973 
0.404 602 
.404 232 
.403 862 

•403 492 
.403 122 

0.402753 
.402 384 
.402015 
.401 646 
.401 278 

0.400 909 
.400 541 
.400173 
•399 806 
•399 438 

0.399071 
•398 7°4 
■39^337 
•397 971 
•397 605 

o-397 239 
•396873 
.396 507 
.396 142 
•395 777 

0.395412 

•395 °47 
•394 683 
.394 318 
•393 954 
Q-393 59o 
Tan. 



60 
59 
58 
57 
56 

55 
54 
53 
52 
51 
5o 
49 
48 
47 
46 

45 
44 
43 
42 

4i 
40 
39 
38 
37 
36 

35 
34 
33 
32 
3i 
30 
29 
28 

27 
26 

25 
24 
23 
22 
21 



19 
18 

17 
16 

15 
14 
13 
12 
11 
10 

9 
8 

7 
6 

5 
4 
3 
2 

1 
o 

m7 



202 LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 

22° 



o 

X 
2 

3 
4 

5 
6 

7 
8 

9 
io 
ii 

12 

13 
H 

15 
16 

17 
18 

19 

20 
21 
22 
23 

24 

25 

26 

27 
28 

29 

30 
31 
32 

33 
34 

35 
36 
37 
38 
39 
40 
4i 
42 
43 
44 

45 
46 

47 
48 

49 
50 
5i 
52 
53 
54 

55 
56 
57 
58 
59 
60 



Sin. 



9-573 575 
.573888 
•574 200 
•574 5*2 
•574 824 

9-575 136 
•575 447 
•575 758 
.576069 

•576 379 
9.576 689 
•57 6 999 
•577 309 
.577618 

•577 927 
9.578 236 
•578 545 
.578853 
•579 162 
•579 47° 

9-579 777 
.580 085 

•580 392 
.580 699 
.581 005 

9.581 312 
.581 618 
.581 924 
.582 229 
•582 535 

9.582 840 
•583 145 
.583 449 
•583 754 
.584058 

9.584 361 
•584 665 
.584 968 
.585 272 
085 574 

9.585877 
.586179 
.586 482 
.586 783 
.587085 

9.587 386 
.587 688 
.587989 
.588 289 
.588 590 

9.588 890 
•589 190 
.589 489 

•589 789 
.590 088 

9-59Q 387 
.590 686 
.590 984 
.591 282 
.591 580 

9-591 878 

Cos. 



D. 1". 



5.22 
5.20 
5.20 
5.20 
5.20 
5.18 
5.18 
5.18 
5-17 
5-17 
5- J 7 
5.17 
5-i5 
5-i5 
515 
5-i5 
5-13 



5.i5 
5-!3 
5.12 

5.13 
5.12 
5.12 
5.10 
5.12 
5.10 
5.10 
5.08 
5.10 
5.08 
5.08 

5-°7 
5.08 

5-07 
5-°5 
5-07 
5-o5 
5-07 
5-°3 
5-°5 
5-°3 
5-o5 
5.02 

5-°3 
5.02 

5-°3 

5.02 
5.00 
5.02 
5.00 
5.00 
4.98 
5.00 
4.98 
4.98 
4.98 
4-97 
4-97 
4-97 
4-97 



D. 1". 



Cos. 



9.967 166 
.967115 
.967 064 
.967013 
.966961 

9.966 910 
.966 859 
.966 808 
.966 756 
.966 705 

9.966 653 
.966 602 
.966 550 
.966 499 
.966 447 

9.966 395 
.966 344 
.966 292 
.966 240 
.966 188 

9.966 136 
.966 085 
.966 033 
.965 981 
.965 929 

9.965 876 
.965 824 
.965 772 
.965 720 
.965 668 

9.965 615 

•965 563 
•965 511 

.965 458 
.965 406 

9-965 353 
.965 301 
.965 248 
•965 195 
•965 i43 

9.965 090 

•965 o37 
.964 984 
.964931 
.964 879 
9.964 826 

•964 773 
.964 720 
.964 666 
.964613 
9.964 560 
.964 507 

.964 454 
.964 400 

•964 347 

9.964 294 

.964 240 

.964 187 

-964133 
.964 080 

9.964 026 
Sin, 



D. 1", 



37 



90 



90 
88 
88 
90 
88 
90 
88 
90 

D, 1". 

67° 



Tan. 



9.606410 
.606 773 
.607 137 
.607 500 
.607 863 

9.608 225 
.608 588 
.608 950 
.609 3 1 2 
.609 674 

9.610036 
•610397 
.610 759 
.611 120 
.611 480 

9.61 1 841 
.612 201 
.612 561 
.612 921 
.613281 

9.613 641 
.614000 

•614 359 
.614 718 
.615 077 

9-615 435 
•615 793 
.616 151 
.616 509 
.616867 

9.617 224 
.617 582 
.617939 
.618 295 
.618652 

9.619 008 

.619 364 
.619 720 
.620 076 
.620432 

9.620 787 
.621 142 
.621 497 
.621 852 
.622 207 

9.622 561 
.622 915 
.623 269 
.623 623 
.623 976 

9.624 330 
.624 683 
.625 036 
.625 388 
.625 741 

9.626 093 
.626445 
.626 797 
.627 149 
.627 501 

9.627 852 
Col 



D. 1". 



6.05 
6.07 
6.05 
6.05 
6.03 

6.05 
6.03 
6.03 
6.03 
6.03 
6.02 
6.03 
6.02 
6.00 
6.02 
6.00 
6.00 
6.00 
6.00 
6.00 
5.98 
5-98 
5.98 
5.98 
5.97 
5-97 
5-97 
5-97 
5-97 
5-95 
5-97 
5-95 
5-93 
5-95 
5-93 
5-93 
5-93 
5-93 
5-93 
5-92 
5-92 
5-92 
5-92 
5-92 
5-9Q 
5-90 
5-90 
5-9° 
5.88 

5-9o 
5.88 
5.88 
5.87 
5.88 
5.87 
5.87 
5.87 
5.87 
5.87 
5.85 

D. 1". 



Cot. 




0-393 590 


60 


•393 227 


59 


.392 863 


58 


•392 5°° 


57 


•392 137 


56 


o-39i 775 


55 


.391412 


54 


.391 050 


53 


.390 688 


52 


.390 326 


5i 


0.389 964 


50 


•389 603 


49 


.389 241 


48 


.388 880 


47 


.388 520 


46 


0.388159 


45 


•387 799 


44 


•387 439 


43 


.387 o79 


42 


.386719 


4i 


0.386 359 


40 


.386 000 


39 


.385 641 


38 


.385 282 


37 


•384 923 


36 


0.384 565 


35 


.384 207 


34 


.383 849 


33 


•383 49i 


32 


.383 133 


3i 


0.382 776 


30 


.382418 


29 


.382061 


28 


•381 705 


27 


.381 348 


26 


0.380 992 


25 


.380 636 


24 


.380 280 


23 


•379 924 


22 


•379 568 


21 


.0.379213 


20 


.378858 


19 


•378 5°3 


18 


.378 148 


17 


•377 793 


16 


o-377 439 


15 


.377085 


14 


•376 731 


13 


•376 377 


12 


.376 024 


11 


0.375 670 


10 


.375 317 


9 


•374 964 


8 


.374612 


7 


•374 259 


6 


o.373 907 


5 


•373 555 


4 


-373 203 


3 


•372851 


2 


•372 499 


1 


0.372 148 




M. 


Tan. 



LOGARITHMIC SINES, COSINES, TANGENTS, 

23° 



AND COTANGENTS. 203 



Sin. 



o 


9.591 878 


I 


.592176 


2 


•592 473 


3 


.592 770 


4 


.593 067 


5 


9-593 363 


6 


•593 659 


7 


•593 955 


8 


•594 251 


9 


•594 547 


IO 


9.594 842 


ii 


•595 137 


12 


•595 432 


13 


•595 727 


14 


.596021 


15 


9-596 3I5 


16 


.596 609 


17 


•596 903 


18 


•597 J 96 


19 


•597 490 


20 


9-597 783 


21 


.598075 


22 


.598 368 


23 


.598 660 


24 


•598 95 2 


25 


9-599 244 


26 


•599 536 


27 


•599 827 


28 


.600 118 


29 


.600 409 


30 


9.600 700 


31 


.600 990 


32 


.601 280 


33 


.601 570 


34 


.601 860 


35 


9.602 150 


36 


.602 439 


37 


.602 728 


38 


.603017 


39 


•603 305 


40 


9.603 594 


4i 


.603 882 


42 


.604 1 70 


43 


.604457 


44 


.604 745 


45 


9.605 032 


46 


.605 319 


47 


.605 606 


48 


.605 892 


49 


.606 179 


50 


9.606 465 


51 


.606 751 


52 


.607 036 


53 


.607 322 


54 


.607 607 


55 


9.607 892 


56 


.608 177 


57 


.608 461 


58 


.608 745 


59 


.609 029 


60 


9.609313 



Cos, 



D. 1' 



4-97 
4-95 
4-95 
4-95 
4-93 
4-93 
4-93 
4-93 
4-93 
4.92 

4.92 
4.92 
4.92 
4.90 
4.90 
4.90 
4.90 
4.88 
4.90 
4.88 
4.87 
4.88 
4.87 
4.87 
4.87 
4.87 
4.85 
4.85 
4.85 
4.85 

4-83 
4.83 
4.83 
4-83 
4.83 
4.82 
4.82 
4.82 
4.80 
4.82 
4.80 
4.80 
4.78 
4.80 

4.78 
478 
478 
4-77 
4.78 

4-77 
4-77 
4-75 
4-77 
4-75 
4-75 
4-75 
4-73 
4-73 
4-73 
4-73 

D. 1". 



Cos, 



9.964 026 
.963972 
.963919 
.963 865 
.963811 

9-963 757 
.963 704 
.963 650 
•963 596 
•963 542 

9.963 488 
.963 434 
•963 379 
•963 325 
.963271 

9.963217 
.963 163 
.963 108 

•963 054 
.962 999 

9.962 945 
.962 890 

-.962836 
.962 781 
.962 727 

9.962 672 
.962 617 
.962 562 
.962 508 
.962 453 

9.962 398 
.962 343 
.962 288 
.962 233 
.962 178 

9.962 123 
.962067 
.962012 
.961 957 
.961 902 

9.961 846 
.961 791 

.961 735 
.961 680 
.961 624 

9.961 569 
.961 513 
.961 458 
.961 402 
.961 346 

9.961 290 
.961 235 
.961 179 
.961 123 
.961 067 

9.961 on 

•960955 
.960 899 
.960 843 
.960 786 
9.960 730 



D, 1". 



D. 1". 

~6& 



Tan. 



9.627 852 
.628 203 
.628 554 
.628 905 
.629 255 

9.629 606 
.629956 
.630 306 
.630 656 
.631 005 

9-631 355 
.631 704 
•632053 
.632 402 
.632 75° 

9-633 o99 
•633 447 
•633 795 
•634 143 
•634 490 

9.634838 
•635 185 
•635 532 
•635 879 
.636 226 

9.636572 
.636919 
.637 265 
.637611 
•637 956 

9.638 302 
.638 647 
•638 992 
•639 337 
.639 682 

9.640 027 
.640371 
.640 716 
.641 060 
.641 404 

9.641 747 
.642091 
.642 434 
•642 777 
.643 120 

9643 463 
.643 806 
.644 148 

•644 490 
.644 832 

9.645 174 
.645 516 

•645 857 
.646 199 
.646 540 
9.646881 
.647 222 
.647 562 

•647 903 
.648 243 

9-648 583 

Cot. 



D. 1", 



5.85 
5.85 
5-85 
5.83 
5-85 
5-83 
5-83 
5.83 
5.82 

5-83 
5.82 
5.82 
5.82 
5.80 
5.82 
5.80 
5.80 
5.80 
5-78 
5.80 

5.78 
5.78 
5-78 
5.78 
5-77 
5.78 
5-77 
5-77 
5-75 
5-77 
5-75 
5-75 
5-75 
5-75 
5-75 
5-73 
5-75 
5-73 
5-73 
5-72 

5-73 

5-72 
5-72 
5-72 
5-72 
5-72 
5-7° 
5-7o 
5-7° 
5-7o 

5-7o 
5.68 

5-7° 
5.68 
5-68 
5.68 

5-67 
5.68 

5-6 7 
5- 6 7 

D. 1", 



Cot. 



0.372 148 
.371 797 
37 l 446 
37i 095 
37o 745 

370 394 
370 044 

369 694 
369 344 
368 995 
368 645 
368 296 
367 947 
367 598 
367 250 
366 901 
366 553 
366 205 

365857 
365 5 IQ 
365 162 
364815 
364 468 
364 121 
363 774 
363 428 
363081 
362 735 
362 389 
362044 

361 698 

361 353 
361 008 
360 663 
360318 

359 973 

359 629 
359 284 
358 940 
358 596 
358 253 
357 909 
357 566 
357 223 
356880 

356 537 
356 194 
355 852 
355 5io 
355 168 
354 826 
354 484 
354 143 
353 801 
353 460 

353 "9 

352 778 

352 438 
352097 
35' 757 

35i 4i7 
Tan. 



60 
59 
58 
57 
56 

55 

54 
53 
52 
5i 
50 
49 
48 
47 
46 

45 
44 
43 
42 

4i 
40 

39 
38 
37 
36 

35 
34 
33 
32 
3i 
30 
29 
28 
27 
26 

25 

24 
23 
22 
21 



19 
18 

17 
16 

15 
14 
13 
12 
n 



204 LOGARITHMIC SINES, 



COSINES, TANGENTS, AND COTANGENTS. 
24° 



Sin. 






9.609313 


I 


.609 597 


2 


.609 8S0 


3 


.610 164 


4 


.610447 


5 


9.610 729 


6 


.611 012 


7 


.611 294 


8 


.611 576 


9 


.611 858 


10 


9.612 140 


ii 


.612421 


12 


.612 702 


13 


.612983 


14 


.613264 


15 


9-6i3 545 


16 


.613825 


17 


.614 105 


18 


.614385 


19 


.614665 


20 


9.614 944 


21 


.615 223 


22 


.615 502 


23 


.615 781 


24 


.616060 


25 


9.616338 


26 


.616616 


27 


.616894 


28 


.617 172 


29 


.617450 


30 


9.617727 


3i 


.618004 


32 


.618281 


33 


.618558 


34 


.618834 


35 


9.619 no 


36 


.619 386 


37 


.619 662 


38 


.619938 


39 


.620213 


40 


9.620488 


41 


.620 763 


42 


.621 038 


43 


.621313 


44 


.621 587 


45 


9.621 861 


46 


.622 135 


47 


.622 409 


48 


.622 682 


49 


.622 956 


50 


9.623 229 


5i 


.623 502 


52 


.623 774 


53 


.624047 


54 


.624319 


55 


9.624 591 


56 


.624 863 


57 


•625 135 


58 


.625 406 


59 


.625 677 


60 


9.625 948 



Cos. 



D. 1' 



4-73 
4.72 

4-73 
4.72 
4.70 
4.72 
4.70 
4.70 
4.70 
4.70 
4.68 
4.68 
4.68 
4.68 
4.68 
4.67 
4.67 
4.67 
4.67 
4.65 
4.65 
4-65 
4-65 
4.65 
4-63 
4-63 
4-63 
4-63 
4.63 
4.62 

4.62 
4.62 
4.62 
4.60 
4.60 
4.60 
4.60 
4.60 
4-58 
4.58 

4-58 
4.58 
4-58 
4-57 
4-57 
4-57 
4-57 
4-55 
4-57 
4-55 
4-55 
4-53 
4-55 
4-53 
4-53 
4-53 
4-53 
4-5 2 
4-5 2 
4-52 

D. 1". 



Cos. 



D. 1". 



9.960 730 
.960 674 
.960618 
.960 561 
.960 505 

9.960 448 
.960 392 
.960 335 
.960 279 
.960 222 

9.960 165 
.960 109 
.960052 
•959 995 
•959 938 

9.959 882 
•959 825 
.959 768 
•959 7 11 
•959 654 

9-959 596 
•959 539 
•959 482 
.959 425 
•959 3 6 8 

9.959 310 
•959 253 
•959 195 
•959 138 
.959 080 

9-959 023 
•958 965 
•958 908 
•958 850 
.958 /92 

9-958 734 
.958 677 
.958 619 
.958 561 
•958 5°3 

9-958 445 
.958 387 
.958329 
.958271 
.958213 

9-958 154 
.958 096 
.958038 
•957 979 
•957 921 

9.957863 
.957 804 

•957 746 
.957 687 
.957628 

9-957 57° 
.957 511 
.957 452 
•957 393 
•957 335 

9-957 276 
Sin. 



•93 
•93 
•95 
•93 
•95 
•93 
•95 
•93 
•95 
•95 
•93 
•95 
•95 
•95 
•93 
•95 
•95 
•95 
•95 
•97 
•95 
•95 
•95 
•95 
•97 
•95 
•97 
•95 
•97 
•95 
•97 
•95 
•97 
•97 
•97 
•95 
-97 
•97 
•97 
•97 
•97 
•97 
•97 
•97 
.98 

•97 
•97 
.98 

•97 
•97 
.98 

•97 



•97 



•98 

•97 
.98 

D. 1" 

~e5° 



Tan. 



9.648 583 
.648 923 
.649 263 
.649 602 
.649 942 

9.650 281 
.650 620 

•650959 
.651 297 
.651636 

9.651 974 
.652312 
.652 650 
.652988 
.653 326 

9.653 663 
.654 000 

•654 337 
.654674 
.655011 

9-655 348 
•655 684 
.656020 
.656356 
.656692 

9.657028 

.657 364 
.657 699 
.658034 
.658369 

9.658 704 
.659 039 
•659 373 
.659 708 
.660 042 

9.660 376 
.660 710 
.661 043 

.661 zn 
.661 710. 

9.662 043 
.662 376 
.662 709 
.663 042 
.663 375 

9.663 707 
.664 039 
.664371 
.664 703. 
.665 035 

9.665 366 
.665 698 
.666029 

.666 360 
.666691 

9.667021 
.667352 
.667 682 
.668013 
.668343 

9-668 673 

Cot. 



D. 1". 



5-67 
5- 6 7 
5.65 

5-67 
5-65 
5-65 
5-65 
5-63 
5.65 

5-63 
5-63 
5-63 
5-63 
5-63 
5.62 

5.62 
5.62 
5.62 
5.62 
5.62 
5.60 
5.60 
5.60 
5.60 
5.60 
5.60 
5.58 
5.58 
5.58 

5-58 
5-58 
5-57 
5.58 

5-57 
5-57 
5-57 
5-55 
5-57 
5-55 
5-55 
5-55 
5-55 
5-55 
5-55 
5-53 
5-53 
5-53 
5-53 
5-53 
5-52 

5-53 
5-52 
5-52 
5-52 
5-5° 
5-5 2 
5-5o 
5-52 
5-5o 
5-5o 

D. 1". 



Cot. 



0.351 417 
•35i 077 
•35° 737 
•35° 398 
.350058 

0.349 719 
•349 380 
•349 041 
•348 7°3 
.348 364 

0.348 026 
.347 688 

•347 35° 
.347012 
.346 674 

0.346 337 
.346 000 
•345 663 
•345 326 
•344 989 

0.344652 
.344 316 
•343 980 
•343 644 
•343 3°8 

0.342 972 
.342 636 
.342 301 
.341 966 
•341 631 

0.341 296 
.340 961 
.340 627 
.340 292 
•339 958 

0.339 624 
.339 290 

•338 957 
.338 623 
.338 290 

0-337 957 
•337 624 
•337 291 
.336 958 
.336 625 

0.336 293 
•335 96i 
•335 629 
•335 297 
•334 965 

0.334 634 
•334 302 
-33397 1 
•333 640 
•333 309 

o.332 979 
.332 648 

•33231% 
.331 987 
•33^ 657 

Q-33I 327 
Tan. 



LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 205 

25° 



M, 



Sin. 






9.625 948 


I 


.626 219 


2 


.626 490 


3 


.626 760 


4 


.627030 


5 


9.627 300 


6 


.627 570 


7 


.627 840 


8 


.628 109 


9 


.628 378 


IO 


9.628 647 


ii 


.628 916 


12 


.629 185 


13 


.629453 


14 


.629 721 


15 


9.629 989 


16 


.630257 


17 


.630 524 


18 


.630 792 


19 


.631 059 


20 


9.631 326 


21 


.631 593 


22 


.631 859 


23 


.632 125 


24 


.632 392 


25 


9.632 658 


26 


.632 923 


27 


•633 189 


28 


•633 454 


29 


•633 719 


30 


9-633 984 


3i 


•634 249 


32 


.634514 


33 


.634 778 


34 


•635 °42 


35 


9.635 306 


36 


•635 57° 


37 


•635 834 


38 


•636 097 


39 


.636 360 


40 


9.636623 


41 


.636 886 


42 


.637 148 


43 


.637411 


44 


.637 673 


45 


9-637 935 


46 


.638 197 


47 


.638458 


48 


.638 720 


49 


.638981 


5o 


9.639 242 


5i 


.639 5°3 


52 


.639 764 


53 


.640 024 


54 


.640 284 


55 


9.640 544 


56 


.640 804 


57 


.641 064 


58 


.641 324 


59 


.641 583 


60 


9.641 842 



Cos. 



D, 1". 



4-52 
4-52 
4-5° 
4-50 
4-5° 
4-5° 
4-5° 
4.48 
4.48 
448 
4.48 
4.48 
4-47 
4-47 
4-47 
4-47 
4-45 
4-47 
4-45 
4-45 
4-45 
4-43 
4-43 
4-45 
4-43 
4.42 

4-43 
4.42 
4.42 
4.42 
4.42 
4.42 
4.40 
4.40 
4.40 
4.40 
4.40 
4-38 
4-38 
4-38 
4-38 
4-37 
4.38 
4-37 
4-37 
4-37 
4-35 
4-37 
4-35 
4-35 
4-35 
4-35 
4-33 
4-33 
4-33 
4-33 
4-33 
4-33 
4-32 
4-32 

D. 1". 



Cos. 



9.957276 
.957217 
•957^8 
•957 °99 
.957040 

9.956981 
.956 921 
.956 862 
.956 803 
•956 744 

9.956 684 
.956 625 
.956 566 
.956 506 
•956 447 

9-956387 
.956327 
.956268 
.956 208 
.956 148 

9.956089 
.956029 
•955 969 
•955 909 
•955 849 

9-955 789 
•955 729 
•955 669 
•955 609 
•955 548 

9-955 488 
-955 428 
•955 368 
•955 307 
-955 247 

9-955 l8 6 
•955 I26 
-955 o6 5 
•955 °o5 
•954 944 

9.954 883 
.954823 
.954 762 
.954 701 
•954 640 

9-954 579 

.954518 

-954 457 
•954 396 
•954 335 
9-954 274 
-954 213 
•954 152 
•954 090 
.954029 

9.953968 
-953 906 
•953 845 
•953 783 
.953 722 

9-953 660 
Sin. 



D. 1' 



.98 



1. 00 
.98 
.98 
.98 

1. 00 



1. 00 
.98 
1. 00 
1. 00 
.98 
1. 00 
1. 00 
.98 
1. 00 
1. 00 
1. 00 
1. 00 
1. 00 
1. 00 
1. 00 
1. 00 
1.02 
1. 00 
1. 00 
1. 00 
1.02 
1. 00 
1.02 
1. 00 
1.02 
1. 00 
1.02 
1.02 
1. 00 
1.02 
1.02 
1.02 
1.02 
1.02 
1.02 
1.02 
1.02 
1.02 
1.02 
1.02 
1.03 
1.02 
1.02 
1.03 
1.02 
1.03 
1.02 
1.03 

D. 1" 

~64P 



Tan, 



9.668 673 
.669 002 
.669 332 
.669 661 
.669 991 

9.670 320 
.670 649 
.670977 
.671 306 
.671 635 

9.671 963 
.672 291 
.672 619 
.672 947 
.673 274 

9.673 602 
.673929 
•674 257 
•674 5 8 4 
.674911 

9-675 237 
•675 564 
.675 890 
.676 217 
•676 543 

9.676 869 
.677.194 
.677 520 
.677 846 
.678171 

9.678496 
.678821 
.679 146 
.679471 
•679 795 

9.680 120 
.680 444 
.680 768 
.681 092 
.681 416 

9.681 740 
.682 063 
.682 387 
.682 710 
.683 033 

9.683 356 
.683 679 
.684001 
•6S4 324 
.684 646 

9.684 968 
.685 290 
.685612 
.685 934 
.686 255 

9.686577 
.686 898 
.687 219 
.687 540 
.687861 

9.688 182 
Cot. 



D. 1' 



5-4» 

5-5° 
548 

5-5o 
5.48 

548 
547 
548 
548 
547 
547 
547 
547 
545 
547 
545 
547 
545 
545 
543 
545 
543 
545 
543 
543 
542 
543 
543 
542 
542 
542 
542 
542 
5-40 
542 
540 
5-4o 
5-40 
5-4o 
54o 
5-38 
54o 
5-38 
5-38 
5-38 
5.38 
5-37 
5-38 
5-37 
5-37 
5-37 
5-37 
5-37 
5-35 
5-37 
5-35 
5-35 
5-35 
5-35 
5-35 

D. 1". 



Cot. 



o-33i 327 
•330 998 
.330 668 
•330 339 
.330 009 

0.329 680 

•329 35 1 
.329 023 
.328 694 
.328 365 
0.328 037 
.327 709 
.327381 
•327053 
.326 726 

0.326 398 
.326071 

•325 743 
.325416 
.325 089 
0.324 763 
•324 436 
.324110 

•323 783 
•323 457 

323 131 

322 806 
322 480 
322154 
321 829 
321 504 
321 179 
320 854 
320 529 
320 205 
319880 

319 556 
319232 
318908 
318584 
318 260 
317 937 
3*7613 
317290 
316967 
316644 
316 321 

3 r 5 999 
315676 

315 354 
315032 
314 710 

3H388 
314066 

313 745 

313423 
313 102 
312 781 
312 460 
312 139 
311 818 
Tan. 



60 
59 
58 
57 
56 

55 
54 
53 
52 
5i 
50 
49 
48 

47 
46 

45 
44 
43 
42 

4i 
40 
39 
38 
37 
36 

35 
34 
33 
32 
3i 
30 
29 
28 
27 
26 

25 
24 
23 
22 
21 



206 LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 

26° 



Sin. 



o 


9.641 842 


I 


.642 101 


2 


.642 360 


3 


.642618 


4 


.642 877 


5 


9.643 135 


6 


•643 393 


7 


.643 650 


8 


.643 908 


9 


.644 165 


IO 


9.644 423 


ii 


.644 680 


12 


.644 936 


13 


.645 l 93 


14 


.645 45° 


15 


9.645 706 


16 


.645 962 


17 


.646218 


18 


.646 474 


19 


.646 729 


20 


9.646 984 


21 


.647 240 


22 


.647 494 


23 


.647 749 


24 


.648 004 


25 


9.648 258 


26 


.648512 


27 


.648 766 


28 


.649 020 


29 


.649 274 


30 


9.649 527 


3i 


.649 781 


32 


.650034 


33 


.650 287 


34 


•650 539 


35 


9.650 792 


36 


.651 044 


37 


.651 297 


38 


•651 549 


39 


.651 800 


40 


9.652052 


4i 


•652 304 


42 


.652 555 


43 


.652 806 


44 


•653 057 


45 


9.653 308 


46 


.653558 


47 


.653 808 


48 


•654059 


49 


•654 309 


50 


9-654 55 8 


5i 


.654 808 


52 


.655 058 


53 


.655 307 


54 


•655 556 


55 


9.655 805 


56 


.656054 


57 


.656 302 


58 


.656551 


59 


.656 799 


60 


9.657047 



Cos. 



D. 1". 



4.32 
4-32 
4-30 
4-32 
4-30 

4-30 

4.28 
4.30 
4.28 
4-30 
4.28 
4.27 
4.28 
4.28 
4.27 
4.27 
4.27 
4.27 
4-25 
4.25 
4.27 
4.23 
4.25 
4.25 
4.23 
4-23 
4-23 
4-23 
4-23 
4.22 

4-23 
4.22 
4.22 
4.20 
4.22 
4.20 
4.22 
4.20 
4.18 
4.20 
4.20 
4.18 
4.18 
4.18 
4.18 

4 

4 
4 
4 
4 
4 
4 



17 
17 
18 

17 

J 5 

17 

J 7 

4-i5 

4.15 

4.15 

4.15 

4-i3 

4-15 

4-13 

4-13 



D. V 



Cos. 



9.953 660 
•953 599 
•953 537 
•953 475 
.953 413 

9-953 352 
.953 290 
•953 228 
•953 166 
•953 104 

9.953 042 
.952980 
.952918 
•952855 
•952 793 

9-952 73I 
.952 669 
.952 606 

•952 544 
.952481 

9.952419 
.952 356 
•952 294 
.9^2231 
.952 168 

9.952 106 
.952 043 
.951 980 
.951 917 
•95 1 854 

9-95 1 79i 
.951 728 
.951 665 
.951 602 
•95i 539 

9-95*476 
.951.412 

•95 1 349 
.951 286 
.951 222 

9-95 1 159 
.951 096 
.951032 
.950 968 
•95° 905 

9.950 841 

•95° 778 
.950714 
•95° 650 
.950 586 
9.950 522 
•950 458 
•95° 394 
•95° 33o 
.950 266 

9.950 202 
.950 138 
.950074 
.950010 
•949 945 

9.949 881 



D. 1' 



.02 
•03 
■03 
•03 
.02 

•03 
■03 
•03 
•03 
•03 
•03 
•03 
•05 
•03 
•03 
•03 
•05 
■03 
•05 
•03 

•°5 
•03 
■05 
■05 

•03 
•05 
•05 
■05 
•05 
■°5 
•05 
-05 
•05 
•05 
•05 
.07 
■05 
■°5 
.07 
.05 

■05 

.07 
.07 

.05 
.07 

■05 

.07 
.07 
.07 
.07 
.07 
.07 
.07 
.07 
.07 
.07 
.07 
.07 
.oS 
■07 

D. 1". 

63° 



Tan. 



9.688 182 
.688 502 
.688 823 
.689 143 
.689 463 

9.689 783 
.690 103 

.690 423 
.690 742 
.691 062 

9.691 381 
.691 700 
.692019 
.692 338 
.692 656 

9.692975 

.693293 
.693 612 

•693 93° 
.694 248 

9.694 566 
.694 883 
.695 201 
.695518 
.695 836 

9-696153 
.696 470 
.696 787 

•697 io 3 
.697 420 

9.697 736 
.698053 
.698 369 
.698 685 
.699 001 

9.699316 
.699 632 
.699 947 
.700 263 
.700578 

9.700 893 
.701 208 

•701 523 
.701 837 
.702 152 
9.702 466 
.702 781 

.703 095 
.703 409 

•703 722 

9.704036 

.704 35° 
.704 663 
.704 976 
.705 290 

9.705 603 
.705 916 
.706 228 
.706 541 
.706 854 

9.707 166 
Cot. 



D. 1' 



5-33 
5-35 
5-33 
5-33 
5-33 
5-33 
5-33 
5-32 
5-33 
5.32 

5-32 
5-32 
5-32 
5-3o 
5-32 
5-3o 
5-32 



5-3o 
5.30 
5-30 
5.28 
5-30 
5.28 
5-3o 
5.28 

5.28 
5.28 

5-27 
5.28 

5- 2 7 
5.28 
5- 2 7 
5-27 
5.27 
5. 2 5 
5-27 
5.25 
5-27 
5- 2 5 
5.25 
5-25 
5-25 
5-23 
5-25 
5.23 

5.25 
5.23 
5-23 
5.22 

5.23 

5-23 
5.22 
5.22 

5-23 
5.22 

5.22 
5.20 
5.22 

5.22 
5.20 



D. 1". 



Cot. 



0.311818 


60 


.311498 


59 


.311 177 


58 


•310857 


57 


•310537 


56 


0.310 217 


55 


•309 897 


54 


.309 577 


53 


•309 258 


52 


.308 938 


5i 


0.308 619 


50 


.308 300 


49 


.307 981 


48 


.307 662 


47 


•307 344 


46 


0.307 025 


45 


.306 707 


44 


.306 388 


43 


.306 070 


42 


•305 752 


4i 


0-305 434 


40 


•305 117 


39 


•304 799 


38 


.304 482 


37 


.304 164 


36 


0.303 847 


35 


.303 53o 


34 


.303 213 


33 


•302 897 


32 


•302 580 


3i 


0.302 264 


30 


.301 947 


29 


.301 631 


28 


•301 315 


27 


•3°o 999 


26 


0.300 684 


25 


.300 368 


24 


.300 053 


23 


.299 737 


22 


.299 422 


21 


0.299 io 7 


20 


.298 792 


19 


.298477 


18 


.298 163 


17 


.297 848 


16 


0.297 534 


15 


.297 219 


14 


.296 905 


13 


.296 591 


12 


.296 278 


11 


0.295 964 


10 


.295 650 


9 


.295 337 


8 


.295 024 


7 


.294 710 


6 


0.294 397 


5 


.294 084 


4 


.293 772 


3 


.293 459 


2 


.293 146 


1 


0.292 834 






Tan. 



LOGARITHMIC SIXES, 



COSINES, TANGENTS, AND COTANGENTS. 207 
27° 



D. 1". 



9.657047 
.657 295 

•657 542 
.657 790 
.658037 
9.658 284 
.658531 
.658 778 
.659025 
.659 271 

9-659 5 X 7 
.659 763 
.660 009 
.660 255 
.660 501 

9.660 746 
.660 991 
.661 236 
.661 481 
.661 726 

9.661 970 
.662 214 
.662459 
.662 703 
.662 946 

9.663 190 
.663 433 
.663 677 
.663 920 
.664 163 

9.664 406 
.664 648 
.664 891 
.665 133 
.665 375 

9.665 617 
.665 859 
.666 100 
.666 342 
.666 583 

9.666 824 
.667 065 
.667 305 
.667 546 
.667 786 

9.668027 
.668 267 
.668 506 
.668 746 
.668 986 

9.669 225 
.669 464 
.669 703 
.669 942 
.670 181 

9.670419 
.670 658 
.670896 
.671 134 
.671 372 

9.671 609 
Co^ 



4-13 

4.12 

4-13 
4.12 
4.12 
4.12 
4.12 
4.12 
4.10 
4.10 
4.10 
4.10 
4.10 
4.10 
4.08 
4.08 
4.08 
4.08 
4.08 
4.07 
4.07 
4.08 
4.07 

4-05 
4.07 

4-05 
4.07 

4.05 
4-05 
4.05 
4-03 
4-05 
403 
403 
4-03 

4-°3 
4.02 

4-03 

4.02 

4.02 

4.02 

4.00 

4.02 

4.00 

4.02 

4.00 

3-98 

4.00 

4.00 

3-98 

3-98 

3-98 

3-98 

3-98 

3-97 

3-98 

3-97 

3-97 

3-97 

3-95 

D. 1" 



D. 1". 



9.949 881 
.949 816 

•949 75 2 
.949 688 
•949 623 
9-949 558 
•949 494 
•949 429 
•949 364 
•949 3°° 
9-949 235 
•949 17° 
•949 105 
•949 °40 
•948 975 
9.948 910 
.948 845 
.948 780 
.948715 
.948 650 
9.948 584 
.948519 
.948 454 
.948 388 
.948 323 
9.948257 
.948 192 
.948 126 
.948 060 
•947 995 
9.947 929 
•947 863 
•947 797 
•947 73i 
•947 66 5 
9.947 600 
•947 533 
•947 467 
•947 401 
•947 335 
9.947 269 
•947 203 
•947 l 3 6 
•947 °7° 
•947 °°4 
9-946 937 
.946871 
.946 804 
.946 738 
.946671 
9.946 604 
.946 538 
.946471 
.946 404 
.946 337 
9.946 270 
.946 203 
.946 136 
.946 069 
.946 002 

9-945 935 
Sin. 



Tan. 



I.08 

1.07 

I.07 

1.08 

1.08 

1.07 

1.08 

1.08 

1.07 

I.08 

1.08 

I.08 

I.08 

1.08 

1.08 

1.08 

I.08 

1.08 

1.08 

1. 10 

I.08 

1.08 

1. 10 

I.08 

1. 10 

1.08 

1. 10 

1. 10 

1.08 

1. 10 
1. 10 
1. 10 
r.io 
1. 10 
1.08 
1. 12 
1. 10 
1. 10 
1. 10 
1.10 
1. 10 
1. 12 
1. 10 
1. 10 
1. 12 
1. 10 
1. 12 
1. 10 
1. 12 
1. 12 
1. 10 
1. 12 
1. 12 
1. 12 
1. 12 
1. 12 
1. 12 
1. 12 
1. 12 
1. 12 

D. 1". 

6SP 



9.707 166 
.707478 
.707 790 
.708 102 
.708414 

9.708 726 
.709 037 

•7°9 349 
.709 660 
.709971 

9.710 282 

• 7 io 593 
.710904 
.711 215 
.711525 

9.71 1 836 
.712 146 
.712456 
.712 766 
.713076 

9-7 J 3 386 
.713696 
.714005 

•7 I 4 3i4 
.714624 

9-7H933 
.715 242 

.715 551 
.715 860 
.716 168 

9.716477 
.716785 
.717093 
.717401 
.717709 

9.718017 

.718325 
.718633 
.718940 
.719 248 

9-719 555 
.719 862 
.720 169 
.720476 
.720 783 

9.721 089 
.721 396 
.721 702 
.722009 
.722315 

9.722 621 
.722 927 
.723 23 2 
•723 538 
.723 844 

9.724 149 
.724 454 
.724 760 
.725065 
.725 37° 

9.725 674 



D. 1". 



Cot. 



Cot. 



5.20 
5.20 
5.20 
5.20 
5.20 
5.18 
5.20 

iis 

5.18 
5.18 
5.18 
5.18 
5.18 

5-i7 
5.18 

5^7 

5-n 

5-i7 

5-!7 

5-*7 

5.i7 

5-!5 

5^5 

5-i7 

5-J5 

5-*5 

5-J5 

5^5 

5-!3 

5-!5 

5-*3 

5-*3 

5-J3 

5-J3 

5'!3 

S- l 3 

5- l 3 

5.12 

5-13 

5.12 

5.12 

5- 12 
5.12 

5- 12 
5.10 

5 12 
5.10 

5- 12 
5.10 

5.10 

5.08 
5.10 
5.10 
5.08 
5.08 
5.10 
5.08 
5.08 
5-°7 



D. 1". 



0.292834 
.292 522 
.292 210 
.291 898 
.291 586 

0.291 274 
.290 963 
.290651 
.290 340 
.290029 

0.289 7 J 8 
.289 407 
.289 096 
.288 785 
.288475 
0.288 164 
.287 854 
.287 544 
.287 234 
.286 924 
0.286614 
.286 304 
•285 995 
.285 686 
.285 376 
0.285 067 
.284 758 
.284 449 
.284 140 
.283 832 
0.283 523 
.283 215 
.282 907 
.282599 
.282 291 
0.281 983 
.281 675 
.281 367 
.281 060 
.280 752 
0.280 445 
.280138 
.279831 
.279 524 
.279 217 
0.278 911 
.278 604 
.278 298 
.277991 
.277 685 

0.277 379 
.277073 
.276 768 
.276462 
.276 156 

0.275 851 
.275 546 
.275 240 
.274 935 
.274 630 

0.274 326 



Tan. 



60 
59 
58 
57 
56 

55 
54 
53 
52 
5i 
50 
49 
48 

47 
46 

45 

44 
43 
42 

4i 

40 

39 

38 

37 

36 

35 

34 

33 

32 

3i 

30 

29 

28 

27 

26 

25 
24 

23 
22 
21 



19 
18 

17 
16 

15 
14 
13 
12 
11 
10 

9 
8 

7 
6 

5 

4 
3 
2 

1 



208 LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 

28° 



M. 



Sin. 






9.671 609 


I 


.671 847 


2 


.672 084 


3 


.672321 


4 


.672558 


5 


9.672 795 


6 


.673032 


7 


.673 268 


8 


•673 505 


9 


.673 741 


10 


9-673 977 


ii 


.674213 


12 


.674 448 


13 


.674 684 


14 


.674919 


15 


9-675 155 


16 


•675 390 


17 


.675 624 


18 


•675 859 


19 


.676094 


20 


9.676328 


21 


.676 562 


22 


.676 796 


23 


.677 030 


24 


.677 264 


25 


9.677 498 


26 


•677 731 


27 


.677964 


28 


.678 197 


29 


.678 430 


30 


9.678 663 


3i 


•678 895 


32 


.679 128 


33 


.679 360 


34 


.679 592 


35 


9.679 824 


36 


.680056 


37 


.680 288 


38 


.680519 


39 


.680 750 


40 


9.680 982 


41 


.681 213 


42 


.681 443 


43 


.681 674 


44 


.681 905 


45 


9.682 135 


46 


.682 365 


47 


•682 595 


48 


.682 825 


49 


.683055 


50 


9.683 284 


51 


.683514 


52 


•683 743 


53 


.683 972 


54 


.684 201 


55 


9.684 430 


56 


.684 658 


57 


.684 887 


58 


.685115 


59 


.685 343 


60 


9-685 571 



Cos. 



D. 1' 



3-97 
3-95 
3-95 
3-95 
3-95 
3-95 
3-93 
3-95 
3-93 
3-93 

3-93 
3-92 
3-93 
3-92 
3-93 
3-92 
3-90 
3-92 
3-92 
3-90 
3.90 
3-90 
3-90 
3-9o 
3-9o 
3.88 
3-88 
3-88 
3-88 
3-88 

3-87 
3-88 
3-8 7 
3-87 
3-8 7 
3-8 7 
3-87 
3.85 
3-8 5 
3-87 
3-8 5 
3-83 
3-85 
3-85 
3.83 
3-^3 
3-83 
3.83 
3-83 
3.82 

3-83 
3-82 
3-82 
3.82 
3.82 
3.80 
3.82 
3.80 
3.80 
3.80 

D. 1". 



Cos. 



9-945 935 
.945 868 
.945 800 

•945 733 
.945 666 

9-945 598 
•945 53i 
.945 464 

•945 396 
•945 3 2 8 

9.945 261 
•945 l 93 
•945 I2 5 
•945 °5 8 
•944 99° 

9.944 922 
•944 854 
•944 786 
•944 7 l8 
•944 650 

9.944 582 
•944 5 H 
•944 446 
•944 377 
•944 309 

9-944 241 
.944172 

•944 104 
•944 036 
•943 967 
9-943 899 
•943 830 
•943 761 
•943 693 
•943 624 

9-943 555 
.943 486 

•943 417 
•943 348 
•943 279 
9.943210 
•943 Hi 
•943 072 
•943 003 
.942 934 

9.942 864 

•942 795 
.942 726 
.942 656 
.942 587 
9.942517 
.942 448 
.942 378 
•942 308 
.942 239 

9.942169 
.942 099 
.942 029 

•94i 959 
.941 889 

9-941 819 

Sin. 



D. 1", 



1. 12 

ii3 

1. 12 

1. 12 
I.I3 

1. 12 
1. 12 

i-i3 

i-i3 
1. 12 

113 
i-i3 

1. 12 

i-i3 
i-i3 

i-i3 

1. 13 

1. 13 
i-i3 
1.13 
113 
113 
JI 5 
1-13 
113 
i-i5 
113 
113 
™S 
1.13 
i-i5 
J-I5 
1. 13 

JI 5 
l -iS 
IJ 5 

115 
i-i5 
i-i5 
i-i5 
1. 15 
115 
i-i5 
i-i5 
1. 17 

IJ 5 

1. 17 

I - 1 5 
1. 17 

115 
1. 17 
1. 17 

115 

1. 17 

1. 17 
1. 17 
1. 17 
1. 17 
1. 17 



D. 1". 

~6P 



Tan. 



9.725 674 

•725 979 
.726 284 
.726 588 
.726 892 

9.727 197 

•727 501 
.727 805 
.728 109 
.728412 

9.728 716 
.729 020 

.729 323 
.729 626 
.729 929 

9-73° 233 
•73o 535 
.730838 

•73i Hi 

•73 1 444 

9.731 746 

•732 048 

•732 351 
•732653 

•732 955 
9733 257 
•733 558 
.733 860 
•734 162 
.734 463 

9-734 764 

.735066 

•735 367 

•735 668 

•735 969 

9.736 269 

•73657° 
.736 870 

.737 171 
•737 471 

9-737 771 
•738071 

•738 371 
.738671 
.738971 

9.739271 
•739 570 
•739 870 
.740 169 
.740 468 

9.740 767 
.741 066 

•741 365 

.741 664 

.741 962 

9.742 261 

.742 559 
.742858 

•743 156 

•743 45" 

9-743 752 

Cot. 



D. 1". 



5.08 
5.08 
5-°7 
5-°7 
5.08 

5-o7 
5-°7 
5-°7 
5-°5 
5-°7 
5-°7 
5-05 
5-o5 
5-05 
5-°7 
5-03 
5-05 
5-05 
5-°5 
5-°3 
5-°3 
5-°5 
5-°3 
5-°3 
5-°3 
5.02 
5-03 
5-°3 
5.02 
5.02 

5-°3 
5.02 
5.02 
5.02 
5.00 
5.02 
5.00 
5.02 
5.00 
5.00 
5.00 
5.00 
5.00 
5.00 
5.00 

4.98 
5.00 
4.98 
4.98 
4.98 
4.98 
4.98 
4.98 
4-97 
4.98 

4-97 
4.98 
4-97 
4.97 
4-97 

D. 1". 



Cot. 



0.274 326 
.274021 
.273716 
.273412 
.273 108 

0.272 803 
.272499 
.272 195 
.271 891 
.271 588 

0.271 284 
.270980 
.270677 
.270 374 
.270071 

0.269 767 
.269 465 
.269 162 
.268 859 
.268556 

0.268 254 
.267 952 
.267 649 
•267 347 
.267 045 

0.266 743 
.266 442 
.266 140 
•265 838 
•265 537 

0.265 2 36 
•264 934 
.264 633 
.264 332 
.264031 

0.263 73i 

.263 430 
.263 130 
.262 829 
.262 529 

0.262 229 
.261 929 
.261 629 
.261 329 
.261 029 

0.260 729 
.260 430 
.260 130 
.259 831 
•259 532 

0.259 233 
.258 934 
.258635 
.258 336 
.258 038 

0.257 739 
.257441 
.257 142 
.256 844 
•256 546 

0.256 248 
Tan. 



LOGARITHMIC SIXES, 



COSINES, TANGENTS, 
29° 



AND COTANGENTS. 209 



M. 


Sin, 


D. 1". 


Cos. 


D. 1". 


Tan. 


D. 1". 


Cot. 







9-685 571 


3-8o 
3.80 
3-78 
3.80 

3-7& 
3-78 
3-78 
3-77 
3-78 
3-78 

3-77 
3-77 
3-77 
3-77 
3-75 
3-77 
3-75 
3-75 
3-75 
3-75 
3-75 
3-75 
3-73 
3-73 
3-73 
3-73 
3-73 
3-73 
3-7 2 
3-73 
3-72 
3-7 2 
3-7 2 
3-7 2 
3.70 


9.941 819 


1. 17 
1.17 

1. 17 

1. 17 
1.17 

1. 18 


9-743 75 2 


4-97 
4-97 
4-95 
4-97 
4-95 
4-97 


0.256 248 


60 


i 


.685 799 


.941 749 


.744050 


•255 95° 


59 


2 


.686 027 


.941 679 


•744 348 


.255 652 


58 


3 

4 


.686 254 
.686 482 


.941 609 

•94i 539 


•744 645 
•744 943 


• 2 55 355 
.255 57 


57 
56 


5 


9.686 709 


9.941 469 


9.745 240 


0.254 760 


55 


6 


.686 936 


.941 398 


1.17 

1. 17 

1. 18 


•745 538 


.254462 


54 


7 


.687 163 


.941 328 


•745 835 


4-95 
4-95 
4-95 
4-95 
4-95 
4-93 
4-95 
4-95 
4-93 


.254 165 


53 


8 


.687 389 


.941 258 


.746 I3 2 


.253868 


52 


9 

10 


.687 616 
9.687 843 


.941 187 
9.941 117 


1. 17 

1. 18 


.746429 
9.746 726 


•253 571 
0.253 274 


5i 
50 


n 


.688 069 


.941 046 


1. 18 


-747 o 2 3 


•252977 


49 


12 


.688 295 


.940 975 




•747 319 


.252 681 


48 


13 


.688 521 


.940 905 


1. 17 

1. 18 


.747 616 


.252 384 


47 


14 


.688 747 


.940 834 


1.18 


•747 913 


.252087 


46 


15 


9.688 972 


9.940 763 




9.748 209 


0.251 791 


45 


16 


.689 198 


.940693 


1. 17 

1. 18 
1. 18 
1. 18 
1. 18 


748 505 


4-93 
4-93 
4-93 
4-93 
4-93 
4-93 
4-93 
4.92 

4-93 
4.92 

4.92 
4.92 
4.92 
4.92 
4.92 
4.92 
4.90 
4.92 
4.90 
4.92 


.251495 


44 


17 
18 


.689 423 
.689 648 


.940 622 
.940551 


.748 801 
•749 097 


.251 199 
.250 903 


43 
42 


19 


.689 873 


.940 480 


•749 393 


.250 607 


4i 


20 


9.690 098 


9.940 409 


1. 18 
1. 18 
1. 18 


9.749 689 


0.250 311 


40 


21 


.690 323 


•940 338 


•749 985 


.250015 


39 


22 


.690 548 


.940 267 


.750281 


•249 7*9 


38 


23 


.690 772 


.940 1 96 


i!i8 
1. 18 


.750576 


.249 424 


37 


24 


.690 996 


.940 125 


.750872 


.249 128 


36 


25 


9.691 220 


9.940 054 


1.20 
1. 18 


9.751 167 


0.248 833 


35 


26 


,691 444 


•939 982 


.751 462 


.248 538 


34 


27 


.691 668 


.939911 


1I18 

1.20 
1. 18 


•751757 


.248 243 


33 


28 


.691 892 


•939 840 


.752052 


.247 948 


32 


29 


.692 115 


•939 768 


•75 2 347 


•247653 


31 


30 


9-692 339 


9-939 697 


1.20 


9.752 642 


0.247 358 


30 


3i 


.692 562 


•939 625 


1. 18 


•75 2 937 


.247 063 


29 


32 


.692 785 


•939 554 


1 .20 


•753 2 3i 


.246 769 


28 


33 


.693 008 


•939 482 


1.20 

1. 18 


-753 5 2 6 


.246 474 


27 


34 


.693 231 


.939410 


.753820 


.246 180 


26 


35 


9-693 453 


3-7 2 
3-70 
3-7o 
3.7o 
3-7° 

3-7° 
3-68 

3-7° 
3-68 
3-68 
3.68 
3-68 
3.68 
367 
3-68 

3-67 
3-67 
3-67 
3-65 
3-67 
3-67 
3-65 
3-65 
3.65 
3-65 


9-939 339 


1.20 


9-754II5 


4.90 
4.90 
4.90 
4.90 
4.90 
4.88 
4.90 
4.88 
4.90 
4.88 


0.245 885 


25 


36 


-.693676 


.939 267 


1.20 


•754 409 


.245 59i 


24 


37 
38 


.693 898 
.694 1 20 


•939 195 
•939 * 2 3 


1.20 
1. 18 


•754 703 
•754 997 


.245 297 
•245 °°3 


23 
22 


39 
40 


.694 342 
9.694 564 


.939052 
9.938 980 


1.20 

1.20 


•755 291 
9-755 585 


.244 709 
0.244415 


21 

20 


4i 


.694 786 


.938 908 


1.20 


•755 878 


.244 122 


19 


42 


.695 007 


.938 836 


1.2.2, 


.756172 


.243 828 


18 


43 


.695 229 


■938 763 


1.20 


.756465 


•243 535 


17 


44 


•695 45° 


.938 691 


1.20 


•756 759 


.243 241 


16 


45 


9.695 671 


9.938619 


1.20 


9-757 °5 2 


4.88 
4.88 
4.88 
4.88 
4.88 


0.242 948 


15 


46 


.695 892 


•938 547 


1.20 


•757 345 


.242 655 


14 


47 


.696113 


•938 475 


1.22 


.757638 


.242 362 


13 


48 


.696 334 


.938 402 


1.20 


•757 931 


.242 069 


12 


49 


.696554 


•938 330 


1.20 


.758 224 


.241 776 


11 


5o 


9.696 775 


9.938 258 


1.22 


9-758 517 


4.88 
4.87 
4.88 
4.87 
4.87 


0.241 483 


10 


5i 


.696 995 


.938 185 


1.20 


.758810 


.241 190 


9 


52 


.697215 


•938 113 


1.22 


•759 102 


.240 898 


8 


53 

54 


.697 435 
.697 654 


.938 040 
•937 967 


1.22 
1.20 


•759 395 
.759687 


.240 605 
.240313 


7 
6 


55 


9.697 874 


9-937 895 


1.22 


9-759 979 


4.88 
4.87 
4.87 
4.87 
4.85 


0.240021 


5 


56 
57 


.698 094 
.698313 


.937 822 
•937 749 


1.22 
1.22 


.760 272 
.760 564 


.239 728 
.239 436 


4 

3 


58 


.698 532 


.937 676 


1.20 


.760 856 


.239 144 


2 


59 


.698751 


-937 604 


1.22 


.761 148 


.238852 


1 


60 


9.698 970 


9-937 53i 




9.761 439 




0.238 561 







Cos. 


D. 1". 


Sin. 


D. 1". 


Cot. 


D. 1". 


Tan, 


M. 



60° 



210 LOGARITHMIC SINES, COSINES, TANGENTS, 

30° 



AND COTANGENTS. 



Sin. 



o 
i 

2 

3 
4 

5 
6 

7 
8 

9 

10 

ii 

12 

13 
14 

15 
16 

17 
i3 

19 

20 
21 

22 
23 

24 

25 
26 

27 
28 

29 

30 

31 
32 

33 
34 
35 
36 
37 
38 
39 
40 

4i 
42 

43 
44 

45 

46 

47 
48 

49 
50 
5i 
52 
53 
54 

55 

56 

57 
53 
59 
60 



D. 1' 



9.698 970 
.699 189 
.699 407 
.699 626 
.699 844 

9.700062 
.700 280 
.700 498 
.700 716 
•7°° 933 

9.701 151 
.701 368 
.701 585 
.701 802 
.702.019 

9.702 236 
.702452 
.702669 
.702 885 
.703 101 

.703 533 

•7°3 749 
.703 964 
.704179 

9-7°4 395 
.704 610 
.704 825 
.705 040 
•7°5 2 54 

9.705469 
.705 683 
.705 898 
.706 112 
.706 326 

9.706 539 
.706 753 
.706967 
.707 180 
•7°7 393 

9.707 606 
.707819 
.708 032 
.708 245 
.708 458 

9.708 670 
.708 882 
.709 094 
.709 306 
.709 518 

9.709 730 
.709 941 

•7 IOI 53 
.710364 

•7io575 

9.710 786 
.710997 
.711 208 
.711419 
.711 629 

9-7 Il8 39 

Co^ 



3-65 

3^3 

3-65 

3-63 

3-63 

3-63 

3-63 

3-63 

3.62 

363 

3.62 

3.62 

3.62 

3.62 

3.62 

3.60 

3.62 

3.60 

3.60 

3.60 

3.60 

3.60 

3-58 

3.58 

3.60 

3.58 
3-58 
3.58 
3-57 
3-58 

3-57 

3-58 

3-57 

3-57 

3-55 

3-57 

3-57 

3-55 

3-55 

3.55 

3.55 

3-55 

3-55 

3-55 

3-53 

3-53 

3-53 

3-53 

3-53 

3-53 

3-5 2 

3-53 
3-5 2 
3-5 2 
3-52 

3-5 2 
3-5 2 
3-5 2 
3-50 
3-5° 

D. 1". 



Cos. 



D. 1' 



9-937 53i 
•937 458 
•937 385 
•937 3i 2 
•937 2 38 

9-937 l6 5 
•937 °9 2 
.937019 
.936 946 
.936 872 

9.936 799 

.936 7 2 5 
.936652 

.936 578 
.936 5°5 

9-936 43 1 
.936 357 
.936 284 
.936 210 
.936 136 

9.936062 
.935 988 
•935 9H 
•935 840 
•935 766 

9-935 69 2 
.935 618 

•935 543 
•935 469 
•935 395 
9-935 3 2 ° 
•935 2 46 
•935 171 
•935 °97 
.935 022 

9.934 948 
•934 873 
•934 798 
•934 7 2 3 
•934 649 

9-934 574 
•934 499 
-934 4 2 4 
•934 349 
•934 2 74 

9-934 199 
•934 123 
.934 048 
-933 973 
•933 898 

9.933 822 
-933 747 
•933 671 
•933 596 
•933 520 

9-933 445 
•933 369 
•933 293 
.933 217 
•933 Hi 
9-933 °66 
Sin. 



Tan. 



1.22 
1.22 
1.22 
I.23 
1.22 
1.22 
1.22 
1.22 
I.23 
1.22 

I.23 
I.22 
1.23 
1.22 
1.23 
1.23 
1.22 
1.23 

1.23 
1.23 

1.23 
1.23 
1.23 

i- 2 3 
1.23 

1.23 
1.25 
1.23 
1.23 
1.25 
1.23 
1.25 
1.23 
1.25 
1.23 
1.25 
1.25 

1.25 
1.23 
1.25 
1.25 

1.25 
1.25 
1.25 
i- 2 5 
1.27 

1.25 
1.25 
1.25 

1.27 
1.25 
1.27 
1.25 
1.27 
1-25 
1.27 
1.27 
1.27 
1.27 
1-25 



D. 1". 



9.761 439 
.761 731 
.762 023 
.762314 
.762 606 

9.762 897 
.763 188 

.763 479 
.763 770 
.764061 

9-764 35 2 
.764 643 

.764 933 
.765 224 

•765 5H 
9.765 805 
•.766095 
.766 385 
.766675 
.766 965 
9.767255 

•767 545 
.767 834 
.768 124 
^768 414 

9.768 7°3 
.768992 
.769 281 

.769 571 
.769 860 

9.770 148 

•77°437 
.770726 
.771015 
-77 1 303 

9-77 1 59 2 
.771 880 
.772 168 
•77 2 457 
•77 2 745 

9.773 033 
•773 321 
.773 608 
.773 896 
•774 184 

9.774 47 1 
•774 759 
•775 46 
•775 333 
•775 621 

9.775 908 

.776 195 
.776482 
.776 768 
•777 055 

9-777 342 
.777 628 

•777 915 

.778 201 

.778488 

9.778 774 



D. 1". 

59° 



Cot. 



4.87 
4.87 
4.85 
4.87 
4.85 

4.85 
4.85 
4.85 

4.85 
4.85 

4.85 

4.83 
4.85 

4-83 
4.85 
4.83 
4.83 
4.83 
4-83 
4.83 

4.83 
4.82 

4-83 
4.83 
4.82 

4.82 
4.82 

4.83 

4.82 

4.80 

4.82 

4.82 

4.82 

4.80 

4.82 

4.80 

4.80 

4.82 

4.80 

4.80 

4.80 

4.78 

4.80 

4.80 

4.78 

4.80 

4.78 

4.78 

4.80 

4.78 

4.78 

4.78 

4-77 

4.78 

4.78 

4-77 
4.78 
4-77 
4.78 
4-77 



Cot. 



0.238 561 
.238 269 

.237 977 
.237 686 

•237 394 
0.237 103 
.236812 
.236521 
.236 230 
.235 939 
0.235 648 

•235 357 
.235 067 
.234 776 
.234 486 

0.234 195 
.233 905 
.233615 
.233 325 
•233035 

0.232 745 

•232455 
.232 166 
.231 876 
.231 586 
0.231 297 
.231 008 
.230719 
.230 429 
.230 140 

0.229 852 
.229 563 
.229 274 
.228985 
.228 697 

0.228 408 
.228 120 
.227 832 
.227 543 
.227 255 

0.226 967 
.226 679 
.226 392 
.226 104 
.225816 

0.225 529 
.225 241 
.224954 
.224 667 
.224 379 

0.224092 
.223 805 
.223518 
.223 232 
.222 945 

0.222 658 
.222 372 
.222085 
.221 799 
.221 512 

0.221 226 



D. 1". 



60 
59 
58 
57 
56 

55 
54 
53 
5 2 
5i 
50 
49 
48 
47 
46 

45 
44 
43 
42 

4i 
40 
39 
38 
37 
36 

35 
34 
33 
32 
3i 

30 
29 
28 
27 
26 

25 
24 

23 
22 
21 
20 

19 
18 

17 
16 

15 
14 
13 
12 
11 
10 

9 
8 

7 
6 

5 
4 
3 
2 

1 



Tan. 



o 

m7~| 



LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 211 

31° 



u 



o 

i 

2 

3 
4 

5 
6 

7 
8 

9 

10 

ii 

12 

13 
14 

15 

16 

I? 
18 



27 

28 

29 

30 
31 
32 

33 
34 

35 
36 
37 
38 
39 
40 

4i 
42 

43 
44 

45 
46 

47 
48 

49 
50 
5i 
52 
53 
54 
55 
56 
57 
58 
59 
60 



Sin, 



9.71 1 839 
.712050 
.712 260 
.712469 
.712679 

9.712889 
.713098 
■7 l 3 308 
•7 I 35 I 7 
.713726 

9-7I3 935 
.714144 

•7H 35 2 
.714561 
.714769 
9.714978 
.715 186 

•7*5 394 
.715 602 
.715 809 
9.716017 
.716 224 
.716432 
.716 639 
.716846 

9717053 
.717259 
.717 466 
.717673 
.717879 

9.718085 
.718 291 
.718497 
.718703 
.718 909 

9.719 114 
.719320 

•7 I 9 5 2 5 
.719730 

•719 935 

9.720 140 

•7 2 ° 345 
.720 549 
.720 754 
.720 958 

9.721 162 
.721 366 
.721 570 
.721 774 
.721 978 

9.722 181 
.722385 
.722588 
.722 791 
.722994 

9.723 197 
.723 400 
.723 603 
.723 805 
.724007 

9.724 210 
Cos. 



D. 1". 



3-5 2 
3-5° 
348 
3-5o 
3-5° 
348 
3-5° 
348 
348 
348 
348 
347 
348 
347 
348 

347 
347 
347 
345 
347 
345 
347 
345 
345 
345 
343 
345 
345 
343 
343 
343 
343 
343 
343 
342 

343 
342 
342 
342 
342 
342 
340 
342 
340 
340 

340 
340 
340 
340 
3.38 
340 
3-38 
3.38 
3.38 
3-38 
3-38 
3-38 
3-37 
3-37 
3-38 

D. 1". 



Cos. 



9.933066 
.932 990 
.932914 
.932 838 
.932 762 

9.932 685 
.932 609 
•932 533 
•932 457 
•932 380 

9-932 304 
.932 228 
.932151 
•932075 
.931 998 

9.931 921 
.931 845 
.931 768 
.931 691 
.931 614 

9-931 537 
.931 460 

•93i 383 
.931 306 
.931 229 

9.931 152 

•93i 075 
.930 998 
.930 921 
•93° 843 

9.930 766 
.930 688 
.930 611 
•93o 533 
•93° 45 6 

9-930 378 
.930 300 
.930 223 

•93o 145 
.930067 

9.929 989 
.929 911 
•929 833 
•929 755 
.929677 

9.929 599 
.929 521 
.929 442 
.929 364 
.929 286 

9.929 207 
.929 129 
.929 050 
.928 972 
.928 893 

9.928 815 
.928 736 
.928657 
.928578 
.928 499 

9.928 420 
Sin\~ 



D. 1' 



SO 
30 
3° 
30 

30 

32 
30 
30 
,32 

30 
32 
30 
32 
30 
32 
32 
32 
32 
32 

D. 1". 

58° 



Tan, 



9.778774 
.779 060 
•779 346 
-779 632 
.779918 

9.780 203 
.780 489 
.780 775 
.781 060 
.781 346 

9.781 631 
.781 916 
.782 201 
.782 486 
.782771 

9.783056 

•783 34i 
.783 626 
.783910 
•784 195 
9.784479 
.784 764 
.785 048 

•785 332 
.785616 

9.785 900 
.786 184 
.786468 
.786752 
.787 036 

9-787 3I9 
.787 603 
.787 886 
.788170 
.788453 

9-788 T£ 
.789019 
.789 302 
.789 585 
.789 868 

9.790 151 

.790 434 
.790 716 
.790999 
.791 281 

9-791 5 6 3 

.791 846 
.792 128 
.792 410 
.792 692 
9.792 974 
.793256 
•793 538 
.793819 
.794 101 

9-794 383 
•794 664 
•794 946 
•795 227 
•795 5° 8 

9-795 789 
Cot. 



D. 1". 



4-77 
4-77 
4-77 
4-77 
4-75 
4-77 
4-77 
4-75 
4-77 
4-75 
4-75 
4-75 
4-75 
4-75 
4-75 
4-75 
4-75 
4-73 
4-75 
4-73 
4-75 
4-73 
4-73 
4-73 
4-73 
4-73 
4-73 
4-73 
4-73 
4.72 

4-73 
4.72 

4-73 
4.72 
4.72 
4.72 
4.72 
4.72 
4.72 
4.72 
4.72 
4.70 
4.72 
4.70 
4.70 
4.72 
4.70 
4.70 
4.70 
4.70 
4.70 
4.70 
4.68 
4.70 
4.70 
4.68 
4.70 
4.68 
4.68 
4.68 

D, 1", 



Cot. 



0.221 226 
.220 940 
.220 654 
.220 368 
.220 082 

0.219 797 
.219511 
.219 225 
.218 940 
.218654 

0.218 369 
.218084 
.217 799 
.217514 
.217 229 

0.216 944 
.216 659 
•216374 
.216090 
.215 805 

0.215 5 21 

.215 236 
.214952 
.214668 
.214 384 

0.214 IO ° 
.213 816 
.213532 
.213 248 
.212 964 

0.212 681 
.212397 
.212 114 
.211 830 
.211 547 

0.21 1 264 
.210981 
.210 698 
.210415 
.210 132 

0.209 849 
.209 566 
.209 284 
.209 001 
.208 719 

0.208437 
.208 154 
.207 872 
.207 590 
.207 308 

0.207 026 
.206 744 
.206462 
.206 181 
.205 899 

0.205 617 
.205 336 
•205 054 
•204 773 
.204492 

0.204 211 



60 
59 
58 
57 
56 

55 
54 
53 
52 
5i 
50 
49 
48 

47 
46 

45 
44 
43 
42 

4i 
40 
39 
38 
37 
36 

35 
34 
33 
32 
3i 
30 
29 
28 
27 
26 

25 
24 

23 
22 
21 
20 

19 
18 

17 
16 

15 
14 
13 
12 
11 



212 LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 



M. 



o 
i 

2 

3 
4 

5 
6 

7 
8 

9 

io 
ii 

12 

13 
14 

15 
16 

17 

iS 

19 

20 
21 

22 
23 

24 

25 
26 

27 

28 
29 

30 
31 
32 

33 
34 

35 
36 
37 
38 
39 
40 

41 
42 

43 
44 

45 
46 

47 
48 

49 
50 
5i 
52 
53 
54 

55 
56 
57 
58 
59 
60 



Sin, 



>»724 210 
.724412 
.724 614 
.724816 
.725017 

1.725 219 
.725 420 
.725 622 
.725 823 
.726024 

1.726 225 
.726426 
.726 626 
.726827 
.727 027 

1.727 228 
.727 428 
.727 628 
.727 828 
.728027 

1.728 227 
.728427 
.728 626 
.728825 
.729024 

1.729 223 
.729422 
.729 621 
.729 820 
.730018 

1.730 217 

•730 4I5 
.730613 
.730811 
.731 009 

1.731 206 
.731 404 
.731 602 

•73i 799 
.731 996 

'•732 193 
•732 390 
•732 587 
.732 784 
.732980 

'•733 177 
•733 373 
•733 569 
•733 765 
•733 96i 

>-734i57 
•734 353 
•734 549 
•734 744 
•734 939 

>-735 135 
•735 33o 
•735 525 
•735 7i9 
•735 914 
^.736 109 



D. 1". 



3-37 
3-37 
3-37 
3-35 
3-37 
3-35 
3-37 
3-35 
3-35 
3-35 
3-35 
3-33 
3-35 
3-33 
3-35 
3-33 
3-33 
3-33 
3-32 
3-33 

3-33 
3-32 
3-32 
3-32 
3-32 
3-32 
3-32 
3-32 
3-3o 
3-32 

3-30 
3-3o 
3-3o 
3-30 
3.28 

3-30 

3-30 
3-28 
3.28 
3.28 
3.28 
3.28 
3.28 

3-27 
3.28 

3-27 
3-27 
3-27 
327 
3-27 
3-27 
3-27 
3-25 
3-25 
3-27 
3-25 
3-25 
323 
3-25 
3-25 

D. 1". 



Cos. 



9.928 420 
.928 342 
.928 263 
.928 183 
.928 104 

9.928 025 
.927 946 
.927 867 
.927 787 
.927 708 

9.927 629 

•927 549 
.927470 
.927 390 
.927310 
9.927 231 

.927 151 
.927071 
.926 991 
.926911 

9.926 831 
.926751 
.926671 
.926 591 
.926511 

9.926431 
.926351 
.926 270 
.926 190 
.926 no 

9.926 029 

•925 949 
.925 868 
.925 788 
.925 707 
9.925 626 
.925 545 
.925 465 
.925 384 

•925 3°3 
9.925 222 
.925 141 
.925 060 
.924 979 
•924 897 
9.924816 

•924 735 
.924 654 

•924 572 
.924491 

9.924 409 

•924 3 2 8 
.924 246 
.924 164 
.924 083 

9.924001 
.923919 
•923 837 
•923 755 
•923 673 

9-923 59i 
Sin. 



D. 1' 



■30 
-32 
■33 
■32 
■32 

•32 
■32 
■33 
■32 
•32 

■33 

■32 
■33 
■33 
■32 

■33 
•33 

■33 
■33 
■33 
■33 
33 
■33 
■33 
■33 
33 
35 
33 
33 
35 
33 
35 
33 
■35 
•35 
•35 
•33 
•35 
■35 
•35 
•35 
•35 
■35 
•37 
■35 
■35 
■35 
•37 
•35 
■37 
■35 
■37 
■37 
■35 
■37 
■37 
37 
37 
'37 
•37 

D, 1". 

57° 



Tan. 



9-795 789 
.796 070 
.796351 
.796 632 
.796913 

9.797 194 
•797 474 
•797 755 
.798036 
.798316 

9.798 596 
.798877 
•799 157 
•799 437 
.799717 

9.799997 
..800277 
.800557 
.800 836 
.801 116 

9.801 396 
.801 675 
.801 955 
.802 234 
.802513 

9.802 792 
.803 072 

•803 35 1 
.803 630 
.803 909 

9.804 187 
.804 466 
.804 745 
.805 023 
.805 302 

9.805 580 
.805 859 
.806 137 
.806415 
.806 693 

9.806971 
.807 249 
.807 C27 
.807 805 
.808 083 

9.808 361 
.808 638 
.808916 
.809 193 
.809471 

9.809 748 
.810025 
.810302 
.810580 
.810857 

9-8ii 134 
.811 410 
.811 687 
.811 964 
.812 241 

9-812517 
Cot. 



D. 1", 



4.68 
4.68 
4.68 
4.68 
4.68 
4.67 
4.68 
4.68 
4.67 
4.67 
4.68 
4.67 
4.67 
4.67 
4.67 
4.67 
4.67 

4-65 
4.67 
4.67 

4-65 
4.67 

4-65 
4-65 
4-65 
4.67 
4-65 
4-65 
4-65 
4-63 
4-65 
4-65 
4-63 
4.65 
4-63 
465 
4-63 
4-63 
4-63 
4-63 
4-63 
4-63 
4.63 
4-63 
4-63 
4.62 

4-63 
4.62 

4-63 
4.62 

4.62 
4.62 

4-63 
4.62 
4.62 
4.60 
4.62 
4.62 
4.62 
4.60 

D. 1". 



Got. 



O.2O4 211 


60 


.203 930 


59 


•203 649 


58 


.203 368 


57 


.203 087 


5<5 


0.202 806 


55 


.202 526 


54 


.202 245 


53 


.201 964 


52 


.201 684 


5i 


0.20I 404 


50 


.201 I23 


49 


.200 843 


48 


.200 563 


47 


.200 283 


46 


0.200 OO3 


45 


.199 723 


44 


•199 443 


43 


.199 164 


42 


.I98884 


4i 


O.I98 604 


40 


.198325 


39 


.198 O45 


38 


.197 766 


37 


.197487 


36 


O.I97 208 


35 


.I96928 


34 


.I96 649 


33 


.I9637O 


32 


.196-091 


3i 


O.I95 813 


30 


•195 534 


29 


.195 255 


28 


.194977 


27 


.194 698 


26 


0.194420 


25 


.194 141 


24 


.193863 


23 


.193 585 


22 


.193 307 


21 


0.193029 


20 


.192751 


19 


.192473 


18 


.192 195 


17 


.191917 


16 


0.191 639 


15 


.191 362 


14 


.191 084 


13 


.190807 


12 


.190529 . 


n 


0.190 252 


10 


.189 975 


9 


.189698 


8 


.189420 


7 


.189 143 


6 


0.188866 


5 


.188590 


4 


.188313 


3 


.188036 


2 


•187 759 


1 


0.187483 






Tan. 



LOGARITHMIC SINES, COSINES, TANGENTS, 

33° 



AND COTANGENTS. 213 



Sin. 



D. 1' 



o 


9.736 109 


I 


•73 6 3°3 


2 


.736498 


3 


.736 692 


4 


.736 886 


5 


9.737 080 


6 


•737 274 


7 


•737 467 


8 


.737661 


9 


•737 855 


IO 


9.738048 


ii 


.738241 


12 


•738 434 


13 


.738627 


14 


.738 820 


15 


9-739 0I3 


16 


•739 206 


17 


•739 398 


18 


•739 590 


19 


•739 783 


20 


9-739 975 


21 


.740 167 


22 


•740 359 


23 


.740 55° 


24 


.740 742 


25 


9.740 934 


26 


•74i 125 


27 


.741316 


28 


.741 508 


29 


.741 699 


30 


9.741 889 


3i 


.742080 


32 


.742 271 


33 


.742 462 


34 


.742 652 


35 


9.742 842 


36 


•743 033 


37 


•743 223 


38 


•743 413 


39 


.743 602 


40 


9-743 792 


41 


.743 982 


42 


.744171 


43 


•744 361 


44 


•744 55° 


45 


9-744 739 


46 


•744 928 


47 


•745 "7 


48 


•745 306 


49 


•745 494 


50 


9.745 683 


5i 


•745 871 


52 


.746 060 


53 


.746 248 


54 


.746 436 


55 


9.746 624 


56 


.746812 


57 


.746 999 


58 


•747 187 


59 


•747 374 


60 


9.747 562 



Cos, 



3-25 

3-23 



3-23 

3-^3 
3.22 

3-23 
323 

3.22 

3.22 
3.22 
3.22 
3.22 
3.22 
3.22 
3.20 
3.20 
3.22 
3.20 
3.20 
3.20 
3.18 
3.20 
3.20 

3.18 
3-i8 
3.20 
3.18 
3-17 

3.18 

3-i8 
3.18 
3-i7 
3-17 
3.18 
3-i7 
3-17 
3-i5 
3-17 
3-i7 
3i5 
3-i7 



3-13 
3.15 
3-13 
3-13 
3*3 

3-*3 
3.12 

3i3 
3.12 

3-13 



D. 1' 



Cos. 



9-923 59I 
.923 509 
.923427 

•923 345 
.923 263 

9.923 181 

.923 098 
.923 016 
•922933 
.922 851 

9.922 768 
.922 686 
.922 603 
.922 520 
.922 438 

9-922 355 

.922 272 
.922 189 
.922 106 
.922 023 

9.921 940 
.921 857 
.921 774 
.921 691 
.921 607 

9.921 524 
.921 441 

•921 357 
.921 274 
.921 190 

9.921 107 
.921 023 
•920 939 
.920 856 
.920 772 

9.920 688 
.920 604 
.920 520 
.920436 
.920352 

9.920 268 
.920 184 
.920 099 
.920015 
.919931 

9.919 846 
.919 762 
.919677 

•919 593 
.919 508 

9.919424 

•919 339 
.919254 
.919 169 
.919085 
9.919000 
.918915 
.918830 

.918745 
.918 659 

9-918574 

Sin, 



D. 1' 



56 c 



Tan. 



9.812 517 
.812 794 
.813070 

.813 347 
.813623 

9.813 899 
.814 176 
.814452 
.814728 
.815 004 

9.815 280 

•815 555 
.815831 
.816 107 
.816382 

9.816658 
.816933 
.817 209 
.817484 
.817759 

9.818035 
.818310 
.818585 
.818860 
.819135 

9.819 410 
.819684 
.819959 
.820 234 
.820 508 

9.820 783 
.821 057 
.821 332 
.821 606 
.821 880 

9.822 154 
.822 429 
.822 703 
.822977 
.823251 

9.823 524 
.823 798 
.824 072 
.824 345 
.824 619 

9.824 893 
.825 166 
•825 439 
.825 713 
.825 986 

9.826 259 
.826532 
.826 805 
.827078 
.827351 

9.827 624 
.827 897 
.828170 
.828 442 
.828715 

9.828 987 
Cot. 



D. 1". 



4.62 
4.60 
4.62 
4.60 
4.60 
4.62 
4.60 
4.60 
4.60 
4.60 

4.58 
4.60 
4.60 
4.58 
4.60 

4.58 
4.60 

4.58 
4.58 
4.60 

4.58 

4-58 
4-58 
4-58 
4.58 

4-57 
4.58 
4.58 
4-57 
4.58 

4-57 
4.58 
4-57 
4-57 
4-57 
4-58 
4-57 
4-57 
4-57 
4-55 
4-57 
4-57 
4-55 
4-57 
4-57 
4-55 
4-55 
4-57 
4-55 
4-55 
4-55 
4-55 
4-55 
4-55 
4-55 
4-55 
4-55 
4-53 
4-55 
4-53 



D, 1". 



Cot. 



0.187 483 
.187 206 
.186930 
.186653 
.186377 

0.186 101 
.185824 
.185 548 
.185 272 
.184 996 

0.184 720 
.184445 
.184 169 
.183893 
.183618 

0.183 342 
.183067 
.182 791 
.182 516 
.182 241 

0.181 965 
.181 690 
.181 415 
.181 140 
.180865 

0.180 590 
.180 316 
.180041 
.179 766 
.179492 

0.179 217 

.178 943 
.178668 
.178394 
.178 120 
0.177 846 

•177 57i 
.177297 
.177023 
.176749 
0.176476 
.176 202 
.175928 

•175 655 
.175 381 

0.175 107 
.174834 
.174561 
.174287 
.174014 

0.173 741 
.173468 

.173 195 
.172 922 
.172649 
0.172376 
.172 103 
.171 830 
.171558 
.171 285 

0.171 013 



Tan. 



60 
59 
58 
57 
56 

55 
54 
53 
52 
5i 
5o 
49 
48 
47 
46 

45 
44 
43 
42 
4i 
40 
39 
38 
37 
36 

35 
34 

33 
32 
3i 
30 
29 
28 
27 
26 

25 
24 
23 
22 
21 



19 
18 

17 
16 

15 
14 
13 
12 
11 
10 

9 
8 



M, 



214 LOGARITHMIC SINES, COSINES, TANGENTS, 

34° 



AND COTANGENTS. 



M. 



o 

i 

2 

3 
4 

5 
6 

7 
8 

9 
io 
ii 

12 

13 
14 

15 
16 

17 
18 

19 

20 
21 
22 
23 

24 

25 
26 

27 
28 

29 
30 
31 
32 

33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 

45 
46 

47 
48 

49 
50 
5i 
52 
53 
54 
55 
56 
57 
58 
59 
60 



Sin. 



9.747 562 
•747 749 
•747 936 
.748123 
.748310 

9.748 497 
.748 683 
.748 870 
.749056 
•749 243 

9.749 429 
.749615 
.749 801 

.749 987 
.750172 

9-75 358 
•75° 543 
.750 729 
.750914 
.751 099 

9.751 284 
.751 469 
•75i 654 
•75 1 8 39 
.752023 

9.752 208 

.752392 

.752 576 
.752 760 

•75 2 944 
9.753128 
•753 312 
•753 495 
•753 679 
.753862 

9.754046 
.754229 
.754412 

•754 595 

•754 778 

9.754 960 

•755 J 43 
•755 326 
•755 508 
•755 690 

9-755 872 
.756054 
.756236 
.756418 
.756 600 

9.756782 
•756963 
•757 144 
•757 326 

•757 507 
9.757688 
.757869 
•758050 
.758 230 
.758411 

9-758 59I 
Cos. 



D. V 



3.12 
3.12 
3.12 
3.12 
3.12 
3.10 
3.12 
3.10 
3.12 
3.10 
3.10 
3.10 
3.10 
3.08 
3.10 
3.08 
3.10 
3.08 
3.08 
3-o8 
3.08 
3.08 
3.08 

3-07 
3.08 

3-07 
3-07 
3-07 
3-07 
3-07 
3-07 
3-05 
3.07 
3-05 
3-07 
3-05 
3-05 
3.05 
3-05 
3-°3 
3-o5 
3-°5 
3-°3 
3-03 
3-°3 
3-03 
3-°3 
303 
3-03 
303 
3.02 
3.02 

3-°3 
3.02 
3.02 
3.02 
3.02 
3.00 
3.02 
3.00 

D, 1". 



Cos. 



9-918574 
.918489 
.918404 
.918318 
.918233 

9.918 147 
.918 062 
.917976 
.917891 
.917805 

9.917 719 
.917634 
.917548 
.917 462 
.917376 

9.917290 
.917204 
.917 118 
.917032 
.916 946 

9.916859 
.916773 
.916 6S7 
.916600 
.916514 

9.916427 
.916341 
.916254 
.916 167 
.916081 

9-9I5 994 
.915907 
.915 820 

•915 733 
.915 646 

9-9I5 559 
.915472 

.915 385 

.915297 

.915 210 

9.915 123 

•915 035 
.914948 
.914860 
•9H773 

9.914685 
.914598 
.914510 
.914 422 
.914 334 

9.914 246 
.914158 
.914070 
.913 982 
.913894 

9.913806 
.913718 
.913630 
.913 541 
•913 453 

9.913 365 
Sin. 



D. 1". 



Tan, 



D. 1". 
"550 



9.828 987 
.829 260 
.829532 
.829 805 
.830077 

9.830 349 
.S30621 

.830893 
.831 165 

•831 437 

9.831 709 
.831 9S1 

.832253 

.832525 
.832 796 

9.833068 

■ 8 33 339 
.833611 
.833 882 
.834 154 

9-834 425 
.834696 
.834967 
.835 238 
.835 5°9 

9.835 780 
.836051 
.836322 
•836 593 
.836 864 

9.837 l 34 
•837 405 
•837 675 
.837 946 
.838216 

9.838487 
.838 757 
.839 027 
.839 297 
.839 568 

9.839 838 
.840 108 
.840 378 
.840 648 
.S40 917 

9.S41 187 

•841 457 
.841 727 
.841 996 
.S42 266 

9-842 535 
.842 805 
.843 074 

-843 343 
.843612 

9.843 882 
.844 I5 1 
.844 420 
.844 689 
.844 958 

9-845 227 
Oot. 



D. 1". 



4-55 
4-53 
4-55 
4-53 
4-53 
4-53 
4-53 
4-53 
4-53 
4-53 
4-53 
4-53 
4-53 
4-5 2 
4-53 
4-5 2 
4-53 
4-5 2 
4-53 
4-5 2 
4-5 2 
4-5 2 
4-52 
4-52 
4-5 2 
4-5 2 
4.52 
4-5 2 
4-52 
4.50 
4-5 2 
4-5° 
4-52 
4-5° 
4-5 2 

4-5° 
4-5° 
4-5° 
4-5 2 
4-5° 
4-5° 
4-5° 
4-5° 
4.48 
4-5° 
4-5° 
4-5° 
4.48 
4-5° 
4.48 

4-5° 
4.48 
4.48 
4.48 

4-5° 
4.48 
4.48 
4.48 
4.48 
4.48 

D. 1". 



Cot, 



0.171 013 


60 


.170 740 


59 


.170468 


58 


.170195 


57 


.169923 


56 


0.169 651 


55 


.169379 


54 


.169 107 


53 


.168835 


52 


.168563 


5i 


0.168 291 


50 


.168019 


49 


.167747 


48 


•167475 


47 


.167 204 


46 


0.166 932 


45 


.166661 


44 


.166389 


43 


.166 118 


42 


.165 846 


4i 


0.165 575 


40 


.165304 


39 


.165 033 


38 


.164 762 


37 


.164491 


36 


0.164220 


35 


.163949 


34 


.163678 


33 


.163407 


32 


.163 136 


3i 


0.162 866 


30 


.162595 


29 


.162325 


28 


.162054 


27 


.161 784 


25 


0.161 513 


25 


.161 243 


24 


.160973 


23 


.160 703 


22 


.160432 


21 


0.160 162 


20 


.159892 


19 


.159 622 


18 


.159 352 


17 


.159083 


16 


0.158 813 


15 


.158543 


14 


.158273 


13 


.158004 


12 


.157 734 


11 


0.157465 


10 


•157 195 


9 


.156 926 


8 


.156657 


7 


.156 3S8 


6 


0.156 118 


5 


.155 849 


4 


.155 58o 


3 


.155 311 


2 


.155042 


1 


O.I54 773 






Tan, 



LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 215 

35° 



Sin, 



D, 1". 



o 


9-758 59I 


I 


.758 772 


2 


•758 952 


3 


•759 132 


4 


.759 312 


5 


9-759 492 


6 


.759672 


7 


•759 852 


8 


.760031 


9 


.760211 


IO 


9.760390 


ii 


.760 569 


12 


.760 748 


13 


.760927 


14 


.761 106 


IS 


9.761 285 


16 


.761 464 


17 


.761 642 


18 


.761 821 


19 


.761 999 


20 


9.762 177 


21 


.762 356 


22 


.762 534 


23 


.762 712 


24 


.762 889 


25 


9.763067 


26 


•763 245 


27 


.763 422 


28 


.763 600 


29 


•7 6 3 777 


30 


9-7 6 3 954 


31 


.764 131 


32 


.764 308 


33 


.764 485 


34 


.764 662 


35 


9.764838 


36 


.765015 


37 


.765 191 


38 


.765 367 


39 


•7 6 5 544 


40 


9.765 720 


41 


.765 896 


42 


.766072 


43 


.766 247 


44 


.766 423 


45 


9.766 598 


46 


.766 774 


47 


.766 949 


48 


.767 124 


49 


.767 300 


50 


9767 475 


5i 


.767 649 


52 


.767 824 


53 


.767 999 


54 


.768173 


55 


9.768348 


56 


.768 522 


57 


.768 697 


58 


.768871 


59 


.769045 


60 


9.769219 


1 


Cos. 



3.02 

3.00 
3.00 

3.00 

3.00 

3.00 

3.00 

2.98 

3.00 

2.98 
2.98 
2.98 
2.98 
2.98 
2.98 
2.98 
2.97 
2.98 
2.97 
2.97 
2.98 
2.97 
2.97 

2.95 

2.97 

2.97 

2.95 

2.97 
2.95 
2.95 

2-95 
2-95 
2-95 
2-95 
2-93 

2-95 
2-93 
2-93 
2-95 
2-93 
2-93 
2-93 
2.92 

2-93 
2.92 

2-93 
2.92 
2.92 

2.93 
2.92 

2.90 
2.92 
2.92 
2.90 
2.92 
2.90 
2.92 
2.90 
2.90 
2.90 

D. 1" 



Cos. 



9.913 365 
.913276 
.913187 
.913099 
.913010 

9.912 922 
.912833 
.912744 
.912655 
.912566 

9.912477 
.912388 
.912 299 
.912 210 
.912 121 

9.912 031 
.911942 
.911853 
.911763 
.911 674 

9.91 1 584 
.911495 
.911405 

•911 315 
.911 226 

9.911 136 
.911 046 
.910 956 
.910 866 
.910 776 

9.910 686 
.910 596 
.910 506 
.910415 
.910325 

9.910 235 
.910 144 
.910054 
.909 963 
.909 873 

9.909 782 
.909 691 
.909 601 
.909 510 
.909419 

9.909 328 
.909 237 
.909 146 

■909 055 
.908 964 

9.908 873 
.908 781 
.908 690 
.908 599 
.908 507 

9.908416 
.908 324 
.908 233 
.908 141 
.908 049 

9-907 958 
Sin. 



D. 1' 



1.48 
1.48 
1.47 
1.48 
1.47 
1.48 
1.48 
1.48 
1.48 
1.48 
1.48 



1.48 
1.50 
1.48 
1.48 
1.50 
1.48 
1.50 
1.48 
1.50 
1.50 
1.48 
1.50 
1.50 
1.50 
1.50 
1.50 
1.50 
1.50 
1.50 
1.52 
1.50 
1.50 
1.52 
1.50 
1.52 
1.50 
1.52 
1.52 
1.50 
1.52 
1.52 
i-5 2 
1.52 

i-5 2 
1.52 
1.52 
1.52 

i-53 
I-5 2 
I-5 2 
1-53 
1.52 

i-53 
l -S 2 
i-53 
i-53 
1.52 

D. 1" 

~54 c 



Tan. 



D. 1". 



9.845 227 

•845 496 
.845 764 
.846 033 
.846 302 

9.846 570 
.846839 
.847 108 

.847 376 
.847 644 

9.847 913 
.848 181 
.848 449 
.848717 
.848 986 

9.849 254 
.849 522 
•849 790 
.850057 

.850 325 

9-850 593 
.850861 
.851 129 
.851 396 
.851 664 

9.851 931 
.852 199 
.852466 

•852 733 
.853001 

9.853 268 

•853 535 
.853802 
.854069 
.854 336 

9.854 603 
.854870 
.855 137 
•855 404 
.855 671 

9-855 938 
.856 204 
.856471 
•856 737 
.857004 

9.857 270 

.857 537 
.857 803 
.858 069 
.858336 

9.858 602 
.858 868 

.859 134 
.859 400 
.859 666 

9.859932 
.860 198 
.860 464 
.860 730 
•860 995 

9.861 261 
Ooti 



4.48 

4-47 
4.48 
448 
4-47 
448 
448 
447 
4-47 
4.48 

4-47 
447 
4-47 
4.48 

4-47 

4-47 
4-47 
4-45 
4-47 
4-47 
4-47 
4-47 
4-45 
4-47 
4-45 
4-47 
4-45 
4-45 
447 
4-45 
4-45 
4-45 
4-45 
4-45 
4-45 
445 
4-45 
4-45 
4-45 
4-45 
4-43 
4-45 
4-43 
4-45 
4-43 

4-45 
4-43 
4-43 
4-45 
4-43 
4-43 
4-43 
4-43 
4-43 
4-43 
4-43 
443 
4-43 
4.42 

4-43 
D. 1" 



Cot. 



O.I54 773 
•154 504 
.154236 

•153 967 
.153698 

0.153 430 
.153161 
.152892 
.152 624 
.152356 

0.152087 
.151 819 

•151 551 

.151283 
.151014 
0.150 746 
.150478 
.150210 

•149 943 
.149675 

0.149407 
.149 139 
.148871 
.148 604 
.148336 

0.148 069 
.147 801 

•147 534 
.147267 
.146999 
0.146 732 
.146465 
.146 198 

.145 931 
.145 664 

O.I45 397 
.145 130 
.144863 
.144596 
.144329 

0.144062 
.143 796 

•i43 5 2 9 
.143263 
.142 996 

0.142 730 
.142463 
.142 197 
.141931 
.141 664 

0.141 398 
.141 132 
.140866 
.140 600 
.140 334 

0.140068 
.139 802 

•139 536 
.139270 
.139005 

o-i38 739 
Tan. 



60 
59 
58 

57 
56 

55 
54 
53 
52 
5i 
50 
49 
48 
47 
46 

45 
44 
43 
42 

4i 
40 
39 
38 
37 
36 
35 
34 
33 
32 
3i 
30 
29 
28 

27 
26 

25 
24 

23 

22 
21 
20 

19 
18 

17 
16 

15 
14 
13 
12 
11 



M. 



216 LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 

36° 



M. 



o 
i 

2 

3 
4 

5 
6 

7 
8 

9 

10 

ii 

12 

13 
14 

15 
16 

17 

18 

19 

20 
21 
22 
23 

24 

25 

26 

27 

28 

29 

30 
31 
32 

33 
34 

35 
36 
37 
38 
39 
40 
4i 
42 
43 
44 

45 
46 

47 
48 

49 
50 
5i 
52 
53 
54 
55 
56 
57 
58 
59 
60 



Sin, 



9- 



.769219 

•7 6 9 393 
.769 566 
.769 740 
.769913 
.770087 
.770 260 

•77° 433 
.770606 
.770 779 
.770952 
.771 125 
.771 298 
.771 470 
.771 643 
•771 815 
.771 987 
.772159 

•772 331 
.772 503 

.772675 
.772 847 
.773018 
.773 190 
•773 361 
•773 533 
•773 7°4 
•773 875 
.774046 
.774217 

774 388 
774 558 
•774 7 2 9 
•774 899 
.775070 

•775 240 
.775410 

•775 58o 
•775 75° 
•775 920 
.776090 
.776259 
.776429 
.776598 
.776 768 

•776 937 
.777 106 

•777 275 
.777 444 
.777613 

•777 78i 
•777 950 
.778119 
.778287 
•778 455 
.778624 
.778792 
.778960 
.779 128 
•779 295 
•779 463 
Cos. 



D. V 



2.90 
2.88 
2.90 
2.88 
2.90 
2.88 
2.88 
2.88 
2.88 
2.88 
2.88 
2.88 
2.87 
2.88 
2.87 
2.87 
2.87 
2.87 
2.87 
2.87 
2.87 
2.85 
2.87 
2.85 
2.87 
2.85 
2.85 
2.85 
2.85 
2.85 
2.83 
2.85 
2.83 
2.85 
2.83 

2.83 
2.83 
2.83 
2.83 
2.83 

2.82 
2.83 
2.82 
2.83 
2.82 
2.82 
2.82 
2.82 
2.82 
2.80 
2.82 
2.82 
2.80 
2.80 
2.82 
2.80 
2.80 
2.80 
2.78 
2.80 

D, l"\ 



Cos. 



9.907 958 
.907 866 
.907 774 
.907 682 
.907 590 

9.907 498 
.907 406 
.907 314 
.907 222 
.907 129 

9.907 037 
.906 945 
.906 852 
.906 760 
.906 667 

9-9o6 575 
.906 482 
.906 389 
.906 296 
.906 204 

9.906 1 1 1 
.906018 
.905 925 
•905 832 
•905 739 

9.905 645 
•905 552 
•905 459 
.905 366 
.905 272 

9.905 179 
.905 085 
.904 992 
.904 898 
.904 804 

9.904 711 
.904 617 

•904 5 2 3 
.904 429 

•904 335 
9.904 241 

•904 i47 
•904053 
.903 959 
.903 864 

9.903 770 
.903 676 
.903 581 
.903 487 
.903 392 

9.903 298 
.903 203 
.903 108 
.903 014 
.902 919 

9.902 824 
.902 729 
.902 634 
.902 539 
.902 444 

9.902 349 



D. 1". 



P. 1". 

53° 



Tan. 



9.861 261 
.861 527 
.861 792 
.862058 
.862 323 

9.862 589 
.862 854 
.863119 
.863 385 
.863 650 

9.863915 
.864180 
.864 445 
.864 710 
.864 975 

9.865 240 
.865 505 
.865 770 
.866 035 
.866 300 

9.866 564 
.866 829 
.867 094 
.867 358 
.867 623 

9.867 887 
.868 152 
.868416 
.868 680 
.868 945 

9.869 209 
.869 473 
.869 737 
.870001 
.870 265 

9.870529 
.870 793 
.871057 
.871 321 
.871 585 

9.871 849 
.872 112 
.872376 
.872 640 
.872 903 

9.873 167 

-^73 43o 
.873 694 

•873 957 
.874 220 

9.874 484 
.874 747 
.875010 
•875 273 
•875 537 

9.875 800 
.876063 
.876 326 
.876 589 
.876852 

9-877 "4 

Cot. 



D. 1' 



443 
4.42 

4-43 
4.42 

4-43 
4.42 
4.42 

4-43 
4.42 
4.42 
4.42 
4.42 
4.42 
4.42 
4.42 
4.42 
4.42 
4.42 
4.42 
4.40 

4-42 
4.42 
4.40 
4.42 
4.40 
4.42 
4.40 
4.40 
4.42 
4.40 
4.40 
4.40 
4.40 
4.40 
4.40 
4.40 
4.40 
4.40 
4.40 
4.40 

4-38 
4.40 
4.40 
4.38 
4.40 

4-38 
4.40 
4-38 
4-38 
4.40 

4-38 
4-38 
4.38 
4.40 

4-38 

4..38 
4-38 
4.38 
4-38 
4-37 

D. 1". 



Cot. 




0.138739 


60 


.138 473 


59 


.138208 


58 


.137942 


57 


.137677 


56 


0.137 411 


55 


.137 146 


54 


.136881 


53 


.136615 


52 


.136350 


5i 


0.136085 


50 


.135 820 


49 


^35 555 


48 


.135 290 


47 


•135025 


46 


0.134760 


45 


•134 495 


44 


.134230 


43 


•133965 


42 


•133700 


4i 


0.133436 


40 


.133171 


39 


.132 906 


38 


.132642 


37 


•132377 


36 


0.132 113 


35 


.131 848 


34 


.131 584 


33 


.131 320 


32 


•131 055 


3i 


0.130 791 


30 


•130527 


29 


.130263 


28 


.129999 


27 


•129735 


26 


0.129 471 


25 


.129 207 


24 


.128943 


23 


.128 679 


22 


.128415 


21 


0.128 151 


20 


.127888 


J 9 


.127 624 


18 


.127360 


17 


.127097 


16 


0.126833 


15 


.126 570 


14 


.126 306 


13 


.126043 


12 


.125 780 


11 


0.125 516 


10 


.125 253 


9 


.124990 


8 


.124727 


7 


.124463 


6 


0.124 2 °° 


5 


•123937 


4 


.123674 


3 


.123411 


2 


.123 148 


1 


0.122886 





Tan. 


M. 



LOGARITHMIC SINES, 



COSINES, TANGENTS, 

37° 



AND COTANGENTS. 217 



H. 



Sin. 



o 

i 

2 

3 

4 

5 

6 

7 
8 

9 
io 
ii 

12 

13 
14 

15 
16 

17 
18 

19 

20 
21 
22 
23 

24 

25 
26 

27 

28 

29 

30 
31 
32 
33 
34 
35 
36 
37 
33 
39 
40 

4i 
42 

43 
44 

45 
46 

47 
48 

49 
50 
5i 
52 
53 
54 
55 
56 
57 
58 
59 
60 



D. 1". 



9-779 463 
.779631 
.779 798 
.779 966 
.78° x 33 

9.780 300 
.780467 
.780 634 
.780 801 
.780 968 

9.781 134 
.781 301 
.781 468 
.781 634 
.781 800 

9.781 966 
.782 132 
.782 298 
.782 464 
.782 630 

9.782 796 
.782 961 
•783127 
.783 292 
.783 458 

9783 623 

.783 788 

•783 953 
.784 118 
.784 282 

9.784447 
.784612 
.784776 
.784941 
.785 105 

9.785 269 
.785 433 
.785 597 
.785 761 

.785 925 
9.786089 
.786252 
.786416 
.786 579 
.786 742 

9.786 906 
.787 069 
.787 232 
•7 8 7 395 
•787 557 

9.787 720 
.787 883 
.788045 
.788 208 
.788 370 

9.788532 
.788 694 
.788856 
.789018 
.789 180 

9-789 342 
Cos. 



2.80 
2.78 
2.80 
2.78 
2.78 
2.78 
2.78 
2.78 
2.78 
2.77 
2.78 
2.78 
2.77 
2.77 
2-77 
2.77 
2.77 
2.77 

2-77 
2.77 

2-75 
2.77 

2-75 
2.77 

2.75 

2-75 

2.75 

2-75 

2-73 

2-75 

2-75 

2-73 

2-75 

2-73 

2-73 

2-73 

2-73 

2-73 

2-73 

2-73 

2.72 

2-73 

2.72 

2.72 

2-73 

2.72 

2.72 

2.72 

2.70 

2.72 

2.72 

2.70 

2.72 

2.70 

2.70 

2.70 

2.70 

2.70 

2.70 

2.70 

D. 1". 



Cos. 



D. 1", 



9.902 349 
.902 253 
.902 158 
.902 063 
.901 967 

9.901 872 
.901 776 
.901 681 
.901 585 
.901 490 

9.901 394 
.901 298 
.901 202 
.901 106 
.901 010 

9.900914 
.900818 
.900 722 
.900 626 
.900 529 
9.900 433 

•900 337 
.900 240 
.900 144 
.900 047 

9.899951 
.899 854 

.899 757 
.899 660 
.899 564 
9.899 467 
•899 37° 
.899 273 
.899 176 
.899078 
9.898981 
.898 884 
.898 787 
.898 689 
.898 592 
9.898 494 
.898 397 
.898 299 
.898 202 
.898 104 
9.898 006 
.897 908 
.897810 
.897712 
.897 614 
9.897516 
.897418 
.897 320 
.897 222 
.897 123 
9.897025 
.896926 
.896 828 
.896 729 
.896631 

9-896 53 2 
Sin. 



Tan. 



1.60 

1.58 

1.58 

1.60 

1.58 

1.60 

1.58 

1.60 

1.58 

1.60 

1.60 

1.60 

1.60 

1.60 

1.60 

1.60 

1.60 

1.60 

1.62 

1.60 
1.60 
1.62 
1.60 
1.62 
1.60 
1.62 
1.62 
1.62 
1.60 
1.62 
1.62 
1.62 
1.62 
1.63 
1.62 

1.62 

1.62 

1.63 

1.62 

1.63 

1.62 

1.63 

1.62 

1.63 

1.63 

i-6 3 

i-6 3 

1.63 

1.63 

1.63 

1.63 

1.63 

1.63 

1.65 

1.63 

1.65 

1.63 

1.65 

1.63 

1.65 



D. 1". 



D. 1". 

~52^ 



9.877 114 
•877377 
.877 640 

•877 903 
.878 165 

9.878428 
.878691 

.878953 
.879 216 
.879 478 

9.879 741 
.880 003 
.880 265 
.880 528 
.880 790 

9.881 052 
.881 314 
.881 577 
.881 839 

.882 101 

9.882 363 
.882625 
.882 887 
.883 148 
.883410 

9.883 672 

.883 934 
.884 196 

.884 457 
.884719 

9.884 980 
.885 242 
.885 504 
.885 765 
.886026 

9.886 288 
.886 549 
.886811 
.887072 
.887 333 

9.887 594 
.887 855 
.888 116 
.888 378 
.888 639 

9.888 900 
.889 161 
.889421 
.889 682 
.889 943 

9.890 204 
.890 465 
.890 725 
.890 986 
.891 247 

9.891 507 
.891 768 
.892 028 
.892 289 
.892 549 

9.892810 

Cot. 



4-38 
4-38 
4-38 
4-37 
4-38 
4.38 
4-37 
4-38 
4-37 
4.38 

4-37 
4-37 
4.38 
4-37 



Cot. 



4-37 

4-37 

4-35 

4-35 

4-37 

4-35 

4-37 

4-35 

4-35 

4-35 

4-35 

4-35 

4-37 

4-35 

4-35 

4-35 

4-33 

4-35 

4-35 

4-35 

4-35 

4-33 

4-35 

4-35 

4-33 

4-35 

4-33 

4-35 

4-33 

4-35 

D. 1". 



4-37 


4.38 


4.37 


4-37 


4-37 


4-37 


4-37 


4-35 


4-37 


4-37 


4-37 


4-37 


4-35 


4-37 


4-35 



0.122886 
.122623 

.122 360 
.I22097 
.121835 

O.I2I 572 
.121 309 
.121 047 
.I20 784 
.120 522 

0.I20 259 
.119997 

•119 735 
.119472 
.119 210 

0.1 1 8 948 
.118686 
.118423 
.118 161 
.117899 

0.117637 

."7 375 
.117113 
.116852 
.116590 
0.116328 
.116066 
.115 804 

.H5 543 
.115 281 

0.1 15 020 
.114758 
.114496 
.114235 
.H3 974 

0.113712 

•H345 1 
.113189 
.112928 
.112 667 

0.1 12 406 
.112145 
.111884 
.111 622 
.111 361 

0.1 1 1 100 
.110839 
.110579 
.110 318 
.110057 

0.109 796 

•109 535 
.109 275 
.109014 
.108753 

0.108493 
.108232 
.107 972 
.107 711 
• 107 451 

0.107 190 



60 
59 
58 
57 
56 

55 
54 
53 
52 
5i 
50 
49 
48 
47 
46 

45 
44 
43 
42 

4i 
40 
39 
38 
37 
36 

35 
34 
33 
32 
3i 

30 
29 
28 
27 
26 

25 
24 
23 
22 
21 



19 
18 

17 
16 

15 
14 
13 
12 
11 
10 

9 
8 

7 
6 

5 
4 
3 
2 

1 



Tan. 



218 LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 

38° 



10 

ii 

12 

13 
14 

15 
16 

17 
18 

19 

20 
21 
22 

23 
24 

25 
26 

27 

28 

29 

30 
31 
32 

33 
34 

35 
36 
37 
38 
39 
40 
41 
42 
43 
44 

45 
46 

47 
48 

49 
5o 
5i 
52 
53 
54 

55 
56 
57 
58 
59 
60 



Sin. 



9.789 342 
.789 504 
.789665 
.789 827 
.789 988 

9.790 149 
.790 310 
.790471 
.790 632 
•79o 793 

9-790 954 
.791 115 
.791 275 
.791 436 
.791 596 

9-791 757 
.791917 
.792077 
.792 237 
•792 397 

9-792 557 
.792 716 
.792876 
•793 035 
•793 195 

9-793 354 
•793 5H 
•793 673 
•793 832 
•793 99i 

9.794 150 
•794 308 
•794 467 
.794 626 

•794 7 8 4 
9.794 942 
•795 IQI 
•795 259 
•795 417 
•795 575 
9-795 733 
•795 891 
.796 049 
.796 206 
.796 364 
9.796 521 
.796 679 
.796836 
•796 993 
•797 15° 

9-797 307 
.797464 
.797 621 
•797 777 
•797 934 

9.798091 
.798 247 
.798 403 
.798 560 
.798 716 

9.798872 



D. 1". 



2.70 
2.68 
2.70 
2.68 
2.68 
2.68 
2.68 
2.68 
2.68 
2.68 
2.68 
2.67 
2.68 
2.67 
2.68 
2.67 
2.67 
2.67 
2.67 
2.67 
2.65 
2.67 
2.65 
2.67 
2.65 
2.67 
2.65 
2.65 
2.65 
2.65 
2.63 
2.65 
2.65 
2.63 
2.63 
2.65 
2.63 
2.63 
2.63 
2.63 
2.63 
2.63 
2.62 
2.63 
2.62 
2.63 
2.62 
2.62 
2.62 
2.62 
2.62 
2.62 
2.60 
2.62 
2.62 
2.60 
2.60 
2.62 
2.60 
2.60 

D. 1"." 



Cos. 



9.896 532 
•896433 
•896 335 
.896 236 
.896 137 

9.896 038 

.895 939 
.895 840 
.895 741 
.895 641 

9-895 542 
•895 443 
•895 343 
•895 244 
•895 145 

9-895 °45 
•894 945 
.894 846 
.894 746 
.894 646 

9.894 546 
.894 446 
•894 346 
.894 246 
.894 146 

9.894046 

•893 946 
.893 846 

•893 745 
•893 645 

9-893 544 
.893 444 

•893 343 
•893 243 
.893 142 

9.893 041 
.892 940 
.892 839 
.892 739 
.892 638 

9.892 536 
•892 435 
•892 334 
.892 233 
.892 132 

9.892 030 
.891 929 
.891 827 
.891 726 
.891 624 

9.891 523 
.891 421 
.891 319 
.891 217 
.891 115 

9.891 013 
.890911 
.890 809 
.890 707 
.890 605 

9-890 5°3 

Sin, 



D. V 



•65 
•63 
.65 

.65 
.65 

.65 
.65 
.65 
.67 
.65 
.65 
.67 
.65 
.65 
.67 
.67 
.65 

•67 
.67 
.67 

•67 
•67 
.67 

.67 
•67 
•67 
■67 
.63 

•67 
.68 

.67 
.68 
.67 
.68 
.68 
.68 
.68 

•67 
.68 

•70 
.68 
.68 
.68 
.68 
.70 
,68 
,70 
,68 
,70 
,68 
,70 
,70 
70 
70 
70 
70 
70 
70 
70 
70 



D. 1". 

~5P 



Tan. 



9.892 810 
.893 070 

.893 331 
.893 591 
.893851 

9.894 in 

•894 372 
.894632 
.894 892 
.895 152 

9.895 412 
.895 672 
.895 932 
.896 192 
.896452 

9.896 712 
.896971 
.897 231 
.897491 
•897 75 1 

9.898010 
.898 270 
.898 530 
.898 789 
.899 049 

9.899 308 
.899 568 
.899 827 
.900 087 
.900 346 

9.900 605 
.900 864 
.901 124 
.901 383 
.901 642 

9.901 901 
.902 160 
.902 420 
.902 679 
.902 938 

9.903 197 

•903 45 6 
.903 714 

-903 973 
.904 232 

9.904491 
.904 750 
.905 008 
.905 267 
.905 526 

9-905 785 
.906 043 
.906 302 
.906 560 
.906 819 

9.907 077 
•907 336 
•907 594 
•907 853 
.908 1 1 1 

9.908 369 
Cot. 



D. 1". 



4-33 
4-35 
4-33 
4-33 
4-33 
4-35 
4-33 
4-33 
4-33 
4-33 
4-33 
4-33 
4-33 
4-33 
4-33 
4-32 
4-33 
4-33 
4-33 
4-3 2 

4-33 
4-33 
4-32 
4-33 
4-32 

4-33 
4-32 
4-33 
4-32 
4-32 
4-32 
4-33 
4-32 
4-32 
4-32 
4-32 
4-33 
4-32 
4-32 
4-32 
4-32 
4-3o 
4-32 
4-32 
4.32 
4-32 
4-3o 
4-32 
4-32 
4-32 
4.30 
4.32 
4-30 
4-32 
4-30 
4-32 
4-30 
4-3 2 
4-30 
4-3° 

D. 1". 



Cot. 



0.107 I 9° 
.106 930 
.106 669 
.106409 
.106 149 

0.105 889 
.105 628 
.105 368 
.105 108 
.104 848 

0.104 588 
.104328 
.104068 
.103 808 
.103 548 

0.103 288 
.103029 
.102 769 
.102 509 
.102 249 

0.101 990 
.101 730 
.101 470 
.101 211 
.100951 

0.100 692 

.100432 
.100173 
.099913 
.099 654 

0.099 395 
.099 136 
.098 876 
.098 617 
.098 358 

0.098 099 
.097 840 
.097 580 
.097 321 
.097 062 

0.096 803 
.096 544 
.096 286 
.096 027 
.095 768 

0.095 509 
.095 250 
.094 992 

•094 733 
.094474 

0.094 215 

•093 957 
.093 698 

•093 440 
.093 181 

0.092 923 
.092 664 
.092 406 
.092 147 
.091 889 

0.091 631 
Tan. 



LOGARITHMIC SINES, COSINES, TANGENTS, 

39° 



AND COTANGENTS. 219 



Sin, 



o 


9.798 872 


I 


.799 028 


2 


•799 184 


3 


•799 339 


4 


•799 495 


5 


9.799651 


6 


.799 806 


7 


•799 962 


8 


.800 117 


9 


.800 272 


10 


9.800 427 


ii 


.800 582 


12 


.800 737 


13 


.800 892 


14 


.801 047 


IS 


9.801 201 


16 


.801 356 


17 


.801 511 


18 


.801 665 


19 


.801 819 


20 


9.801 973 


21 


.802 128 


22 


.802 282 


23 


.802 436 


24 


.802 589 


2S 


9.802 743 


26 


.802 897 


27 


.803 050 


28 


.803 204 


29 


•803 357 


30 


9.803 511 


3i 


.803 664 


32 


.803817 


33 


.803 970 


34 


.804 123 


35 


9.804 276 


36 


.804 428 


37 


.804 581 


38 


.804 734 


39 


.804 886 


40 


9.805 039 


4i 


.805 191 


42 


•805 343 


43 


•805 495 


44 


.805 647 


45 


9.805 799 


46 


•805951 


47 


.806 103 


48 


.806 254 


49 


.806 406 


5o 


9.806 557 


5i 


.806 709 


52 


.806 860 


53 


.807011 


54 


.807 163 


55 


9.807 314 


56 


.807 465 


57 


.807 615 


58 


.807 766 


59 


.807 917 


60 


9.808 067 



Cos. 



D. 1' 



60 

60 
5S 
60 
60 

58 
60 

5S 
5S 
58 

58 

58 
58 
5S 
57 

5? 
58 

57 
57 
57 
58 
57 
57 
55 
57 
57 
55 
57 
55 
57 
55 
55 
55 
55 
55 
53 
55 
55 
53 
55 
53 
53 
53 
53 
53 
53 
53 
52 
53 
5 2 
53 
5 2 
52 
53 
5 2 
52 
5° 
■5 2 
5 2 
50 



D. 1". 



Cos. 



9.890 503 
.890 400 
.890 298 
.890 195 
.890 093 

9.889 990 
.889 888 
.889 785 
.889 682 
.889 579 

9.889 477 
.889 374 
.889271 
.889 168 
.889 064 

9.888961 
.888 858 
.888 755 
.888651 
.888 548 

9.888 444 
.888341 
.888 237 
.888 134 
.888 030 

9.887 926 
.887 822 
.887718 
.887 614 
.887510 

9.887 406 
.887 302 
.887 198 
.887 093 
.886 989 

9.886 885 
.886 780 
.886 676 
.886571 
.886 466 

9.886362 
.886257 
.886 152 
.886 047 
•885 942 

9.885 837 
.885 732 
.885 627 
.885 522 
.885416 

9.885 311 
•885 205 
.885 100 
.884 994 
.884 889 

9-88 4 783 
.884 677 

.884572 
.884 466 
.884 360 

9-884 254 

Sin. 



D. 1' 



Tan. 



D. 1". 

50° 



9.908 369 
.908 628 

.908 886 
.909 144 
.909 402 

9.909 660 
.909 918 
.910177 

.910435 
.910693 

9.910 951 
.911 209 
.911467 
.911725 
.911 982 

9.912 240 
.912498 
.912756 
.913014 
.913271 

9-9I3 5 2 9 
•913 787 
.914044 
.914302 
.914 560 

9-914817 
.915075 
•915 332 
•915 59o 
.915 847 

9.916 104 
.916 362 
.916 619 
.916 877 
•917 134 

9.917 39i 
.917 648 
.917 906 
.918 163 
.918420 

9.918677 
.918934 
.919191 
.919448 
.919 705 

9.919 962 
.920 219 
.920476 
•920 733 
.920 990 

9.921 247 
.921 503 
.921 760 
.922017 
.922 274 

9.922 530 
.922 787 
•923P44 
.923 300 
.923 557 

9923814 
Cot. 



D. 1". 



4-32 
4.30 
4-3° 
4-30 
4-3° 
4-30 
4.32 
4-30 
4-3° 
4-3° 

4-3° 
4-3o 
4-3° 
4.28 

4-30 

4-3° 
4-30 
4-3° 
4.28 

4-3° 

4-3° 
4.28 

4-3° 
4-30 
4.28 

4-3o 
4.28 
4-30 
4.28 
4.28 

4-30 
4.28 
4-30 
4.28 
4.28 
4.28 
4-30 
4.28 
4.28 
4.28 
4.28 
4.28 
4.28 
4.28 
4.28 
4.28 
4.28 
4.28 
4.28 
4.28 
4.27 
4.28 
4.28 
4.28 
4.27 
4.28 
4.28 
4.27 
4.28 
4.28 

D. 1". 



Cot. 



0.091 631 
.091 372 
.091 114 

.090 856 
.090 598 
0.090 340 
.090 082 
.089 823 
.089 565 
•089 307 

0.089 °49 
.088 791 
•088 533 
.088 275 
.088018 

0.087 760 
.087 502 
.087 244 
.086 986 
.086 729 

0.086471 
.086 213 
.085 956 
.085 698 
•085 440 

0.085 l %3 
.084 925 
.084 668 
.084410 
.084 153 

0.083 896 
.083 638 
•083 381 
.083 123 
.082 866 

0.082 609 
.082352 
.082 094 
.081 837 
.081 580 

0.081 323 
.081 066 
.080 809 
.080 552 
.080 295 

0.080 038 
.079 781 

.079 524 
.079 267 
.079010 

0.078 753 
•078 497 
.078 240 

.077 983 
.077 726 

0.077 470 
.077213 
.076956 
.076 700 
.076443 

0.076 186 
Tan. 



60 
59 
58 
57 
56 

55 

54 
53 
52 
5i 
5o 
49 
48 
47 
46 

45 

44 
43 
42 
4i 
40 
39 
38 
37 
36 

35 
34 
33 
32 
3i 
30 
29 
28 
27 
26 

25 
24 
23 
22 
21 



19 

18 

17 
16 

15 
14 
13 
12 
11 



220 LOGARITHMIC SINES, 



COSINES, TANGENTS, 
40° 



AND COTANGENTS. 



Sin. 



D, 1". 



Cos, 



D 1". 



Tan. 



D. 1", 



Cot. 



o 

i 

2 

3 

4 

5 
6 

7 
8 

9 

10 

ii 

12 

13 
14 

15 
16 

17 
18 

19 

20 
21 
22 
23 

24 

25 
26 

27 

28 
29 
30 
31 
32 

33 
34 

35 
36 
37 
38 
39 
40 

41 
42 

43 
44 

45 
46 

47 
48 

49 
5o 
5i 
52 
53 
54 
55 
56 
57 
58 
59 
60 



.808 067 
.808218 
.808 368 
.808519 
.808 669 
.808 819 
.808 969 
.809 119 
.809 269 
.809419 
.809 569 
.809718 
.809 868 



9.8 



9.8 



9.8 



9.8 



9.8 



9.8 



0017 

167 
0316 
0465 
0614 
0763 
0912 

1 061 
1 210 
1358 
1507 
1655 

1 804 

I95 2 

2 100 
2 248 
2396 

2 544 

2 692 
2840 
2988 

3 135 
3283 
3 43o 
3 578 

3 725 
3872 

4019 
4166 

4313 

4460 
4607 

4 753 
4900 
5046 

5 193 
5 339 

5 485 
5632 
5 778 

5 924 
6069 

6215 
6361 
6507 

6 652 
6798 

6 943 



Cos. 



2.52 
2.50 
2.52 
2.50 
2.50 
2.50 
2.50 
2.50 
2.50 
2.50 
2.48 
2.50 
2.48 
2.50 
2.48 
2.48 
2.48 
2.48 
2.48 
2.48 
2.48 
2.47 
2.48 
2.47 
2.48 
2.47 
2.47 
2.47 
2.47 
2.47 
2.47 
2.47 
2.47 

245 

2.47 

2-45 

2.47 
2.45 
2.45 
2.45 
245 
2.45 
2.45 
2.45 
2.43 

2-45 
2-43 
2-45 
243 
2-43 

2-45 
2-43 
2-43 
2.42 

2-43 
2-43 
2-43 

2.42 

2-43 
2.42 



9.884 254 
.884 148 
.884 042 
■883 936 
.883 829 

9-883 723 
.883617 
.883510 
.883 404 
•883 297 

9.883 191 
.883 084 
.882977 
.882871 
.882 764 

9.882657 
.882550 
.882 443 
.882336 
.882 229 

9.882 121 
.882014 
.881 907 
.881 799 
.881 692 

9.881 584 
.881 477 
.881 369 
.881 261 
.881 153 

9.881 046 
.880 938 
.880 830 
.880 722 
.880613 

9.880 505 
.880 397 
.880 289 
.880 180 
.880072 

9.879 963 
.879855 
•879 746 
•879 637 
.879529 

9.879420 
.879311 
.879 202 
.879093 
.878 984 

9.878875 
.878 766 
.878656 
.878 547 
.878438 

9.878 328 
.878219 
.878 109 
.877999 
.877 890 

9-877 780 
Sin. 



D. 1" 

49^ 



9.923814 
.924 070 
.924 327 
.924 583 
.924 840 

9.925 096 

•925 352 
.925 609 
.925 865 
.926 122 
9.926378 
.926 634 
.926 890 
.927147 
.927 403 

9.927 659 
.927915 
.928 171 
.928427 
.928 684 

9.928 940 
.929 196 
.929452 
.929 708 
.929 964 

9.930 220 
•930 475 
•930 731 
.930 987 

.931 243 

9-93 1 499 
.931 755 
.932010 
.932 266 
.932 522 

9.932 778 
•933 033 
•933 289 
•933 545 
•933 800 

9.934056 
.934 311 
•934 5 6 7 
•934 822 
.935 078 

9-935 333 
•935 589 
•935 844 
.936 100 

•936 355 

9.936 611 
.936 866 
.937 121 
•937 377 
•937 6 32 

9.937 887 
.938 142 
.938 398 
•938 653 
.938 908 

9-939 163 
Cot. 



4.27 
4.28 
4.27 
4.28 
4.27 
4.27 
4.28 
4.27 
4.28 
4.27 
4.27 
4.27 
4.28 
4.27 
4.27 
4.27 
4.27 
4.27 
4.28 
4.27 
4.27 

4.27 
4.27 
4.27 
4.27 

4.25 
4.27 
4.27 

4.27 
4.27 

4.27 

4-25 
4.27 
4.27 
4.27 

4-25 
4.27 
4.27 

4-25 
4.27 

4-25 
4.27 

4.25 
4.27 

4.25 
4.27 

4-25 
4.27 

4-25 

4.27 

4-25 
4.25 
4.27 

4-25 
4-25 

4-25 
4.27 

4.25 
4.25 
4.25 

D. 1". 



0.076 186 


60 


•075 930 


59 


•075 673 


58 


•075417 


57 


.075 160 


56 


0.074 904 


55 


.074 648 


54 


.074391 


53 


•074 135 


52 


.073 878 


5i 


0.073 622 


50 


.073 366 


49 


•073 no 


48 


.072 853 


47 


.072 597 


46 


0.072 341 


45 


.072 085 


44 


.071 829 


43 


•071 573 


42 


•071 316 


4i 


0.071 060 


40 


.070 804 


39 


.070 548 


38 


.070 292 


37 


.070 036 


36 


0.069 780 


35 


.069 525 


34 


.069 269 


33 


.069013 


32 


.068 757 


3i 


0.068 501 


30 


.068 245 


29 


.067 990 


28 


•067 734 


27 


.067 478 


26 


0.067 222 


25 


.066 967 


24 


.066 711 


23 


.066455 


22 


.066 200 


21 


0.065 944 


20 


.065 689 


19 


.065 433 


18 


.065 178 


17 


.064 922 


16 


0.064 667 


15 


0064411 


14 


.064 156 


13 


.063 900 


12 


.063 645 


11 


0.063 389 


10 


.063 134 


9 


.062 879 


8 


.062 623 


7 


.062 368 


6 


0.062 113 


5 


.061 858 


4 


.061 602 


3 


.061 347 


2 


.061 092 


1 


0.060 837 





Tan. 


M. 



LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 221 

41° 



o 

i 

2 

3 
4 

5 
6 

7 
8 

9 

10 

ii 

12 

13 
14 

15 
16 

17 
18 

19 

20 
21 
22 
23 

24 

25 

26 

27 

23 
29 
33 

31 

32 

33 
34 

35 
36 

37 
38 

39 
40 

4i 
42 

43 



45 
46 

47 
48 

49 
50 
5i 
52 
53 
54 
55 
56 
57 
58 
59 
60 



Sin. 



9.816943 
.817088 
.817233 
.817 379 
.817524 

9.817668 
.817813 
.817958 
.818 103 
.818247 

9.818392 
.818536 
.818681 
.818825 
.818969 

9.8I9II3 

.819257 
.819401 

.819 545 
.819 689 

9.819832 
.819976 
.820 120 
.820 263 
.820 406 

9.820 550 
.820 693 
.820 836 
.820 979 
.821 122 

9.821 265 
.821 407 
.821 550 
.821 693 
.821 835 

9.821 977 
.822 120 
.822 262 
.822 404 
.822 546 

9.822 688 
.822830 
.822972 
.823114 
.823255 

9823 397 
.823 539 
.823 680 
.823821 
.823 963 

9.824 104 
.824 245 
.824 386 
.824527 
.824668 

9.824 808 
.824 949 
.825 090 
.825 230 
.825 371 

9-825 5" 
Cos. 



D, 1". 



2.42 
2.42 

2-43 
2.42 
2.40 
2.42 
2.42 
2.42 
2.40 
2.42 
2.40 
2.42 
2.40 
2.40 
2.40 
2.40 
2.40 
2.40 
2.40 
2.38 
2.40 
2.40 
2.38 
2.38 
2.40 

2.38 

2.38 
2.38 
2.38 
2.38 

2-37 

2.38 

2.38 

2-37 

2-37 

2.38 

2-37 

2-37 

2-37 

2-37 

2-37 

2-37 

2.37 
2-35 
2-37 
2.37 
2-35 
2-35 
2-37 
2-35 
2-35 
2-35 
2-35 
2-35 
2-33 
2.35 
2.35 
2-33 
2-35 
2-33 

D. 1". 



Cos. 



D. 1". 



9.877 780 
.877670 
.877 560 

•87745° 
.877 340 

9.877 230 
.877 120 
.877010 
.876 899 
.876 789 

9.876678 
.876568 
.876457 
.876347 
.876 236 

9.876 125 
.876014 
.875 904 

.875 793 
.875 682 

9-875 57i 
.875 459 
.875 348 
.875 237 
.875 126 

9.875014 
.874 903 
.874 79i 
.874 680 
.874 568 

9.874456 
.874 344 
.874232 
.874 121 
.874 009 

9.873 896 
.873 784 
.873672 

.873 5 6 ° 
.873 448 

9-873 335 
.873223 
.873110 

.872998 

.872 885 

9.872772 

.872659 

.872547 
.872434 
.872321 

9.872 208 
.872095 
.871 981 
.871 868 
.871 755 

9.871 641 
.871 528 
.871414 
.871 3 QI 
.871 187 

9-87 I Q 73 

Sin. 



Tan. 



1.83 
1.83 
1.83 
1.83 
I.83 
I.83 
1.85 

1.83 
1.85 

I.83 
I.85 
1.83 
1.85 
1.85 
1.85 
1.83 
I.85 
1.85 
I.85 

1.87 
I.85 
I.85 
1.85 
1.87 
I.85 
I.87 
1.85 
1.87 
I.87 
I.87 
I.87 
I.85 
1.87 
1.88 

1.87 
1.87 
1.87 
1.87 
1.88 

1.87 

1. 88 

1.87 

1.88 

1.88 

1.88 

1.87 

1.88 

1.88 

1.8S 

1.88 

1.90 

1.88 

1.88 

1.90 

1.88 

1.90 

1.88 

1.90 

1.90 

~4& 



D. 1", 



9-939 163 
.939418 
•939 673 
•939 928 
.940 183 

9-940 439 
.940 694 
.940 949 
.941 204 
•94i 459 

9.941 713 
.941 968 
.942 223 
•942 478 
•942 733 

9.942 988 
•943 243 
•943 498 
•943 752 
.944007 

9.944 262 
•944 5 : 7 
•944 77 1 
.945 026 

•945 281 
9-945 535 
•945 790 
.946 045 
.946 299 
.946 554 
9.946 808 
.947 063 
•947 3i8 
•947 572 
.947 827 

9.948 081 

.948 335 
.948 590 
.948 844 
•949 099 

9-949 353 
.949 608 
.949 862 
.950 116 
.950371 

9.950 625 
.950 879 

•95 1 : 33 
.951388 
.951 642 

9.951 896 

•952 15° 
.952 405 
.952659 
.95 2 9!3 
9.953 167 
.953 421 
•953 675 
.953 929 
•954 183 

9-954 437 



Cot. 



4-25 
4-25 
4-25 
4-25 

4.27 

4.25 
4-25 
4-25 
4.25 
4.23 
4.25 
4-25 
4-25 
4-25 
4.25 
4-25 
4.25 
4.23 
4-25 
4.25 
4-25 
4-23 
4-25 
4.25 
4-23 

4-25 
4.25 
4-23 
4-25 
4-23 
4.25 
4.25 
4-23 
4.25 

4-23 

4-23 

4-25 

4-23 

4-25 

4-23 

4-25 

4-23 

4.23 

4.25 

4-23 

4-23 

4-23 

4-25 

4-23 

4.23 

4-23 

4.25 

4-23 

4-23 

4-23 

4-23 

4-23 

4-23 

4-23 

4-23 



Cot. 



0.060 837 
.060 582 
.060 327 
.060072 
.059817 

0.059 561 
.059 306 
.059051 
.058 796 
.058 541 

0.058 287 
.058032 
.057 777 
•057 522 
.057 267 

0.057 012 
.056 757 
.056 502 
.056 248 
.055 993 

0.055 738 
.055 483 
.055 229 

.054 974 
.054719 

0.054 465 
.054210 
.053 955 
.053 7 01 
•053 446 

0.053 192 
.052937 
.052682 
.052 428 
.052173 

0.051 919 
.051 665 
.051 410 
.051 156 
.050 901 

0.050 647 
.050 392 
.050 138 
.049 884 
.049 629 

0.049 375 
.049 121 
.048 867 
.048612 
.048 358 

0.048 104 
.047 850 
.047 595 
.047 34i 
.047 087 

0.046 833 
.046579 
.046 3 2 5 
.046071 
.045817 

0.045 563 



60 
59 
58 
57 
56 
55 
54 
53 
52 
5i 
50 
49 
48 
47 
46 

45 
44 
43 
42 

4i 
40 
39 
38 
37 
36 

35 
34 
33 
32 
3i 
30 
29 
28 

27 
26 

25 
24 

23 
22 
21 
20 

19 
18 

17 
16 

15 

14 
13 
12 
11 



D. 1' 



Tan. 



222 LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 

42° 



M, 



Sin. 



o 


9.825 511 


I 


.825651 


2 


.825 791 


3 


.825 931 


4 


.826071 


5 


9.826 211 


6 


.826351 


7 


.826491 


8 


.826631 


9 


.826 770 


IO 


9.826910 


ii 


.827 049 


12 


.827 189 


13 


.827 328 


14 


.827 467 


15 


9.827 606 


16 


•827 745 


17 


.827 884 


18 


.828 023 


19 


.828 162 


20 


9.828 301 


21 


.828 439 


22 


.828 578 


23 


.828 716 


24 


.828 855 


25 


9.828 993 


26 


.829 131 


27 


.829 269 


28 


.829 407 


29 


.829 545 


30 


9.829 683 


31 


.829821 


32 


.829 959 


33 


.830 097 


34 


.830 234 


35 


9.830 372 


36 


.830 509 


37 


.830 646 


38 


.830 784 


39 


.830921 


40 


9.831 058 


41 


•831 195 


42 


.831 332 


43 


.831 469 


44 


.831 606 


45 


9.831 742 


46 


.831 879 


47 


.832015 


48 


.832 152 


49 


.832 288 


50 


9.832425 


5i 


.832 561 


52 


.832 697 


53 


.832 833 


54 


.832 969 


55 


9-833 105 


56 


.833 241 


57 


•833 377 


58 


•833 5 12 


59 


.833 648 


60 


9-833 783 



Cos. 



D. 1' 



2.33 
2-33 
2-33 
2.33 
2-33 
2-33 
2-33 
2-33 
2.32 

2.33 
2.32 

2-33 
2.32 
2.32 
2.32 
2.32 
2.32 
2.32 
2.32 
2.32 
2.30 
2.32 
2.30 
2.32 
2.30 
2.30 
2.30 
2.30 
2.30 
2.30 
2.30 
2.30 
2.30 
2.28 
2.30 
2.28 
2.28 
2.30 
2.28 
2.28 
2.28 
2.28 
2.28 
2.28 
2.27 
2.28 
2.27 
2.28 
2.27 
2.28 
2.27 
2.27 
2.27 
2.27 
2.27 
2.27 
2.27 
2.25 
2.27 
2.25 

D. 1". 



Cos. 



9.871 073 
.870 960 
.870 846 
.870732 
.870618 

9.870 504 
.870 390 
.870 276 
.870 161 
.870 047 

9-869 933 
.869818 
.869 704 
.869 589 
.869 474 

9.869 360 
.869 245 
.869 130 
.869015 
.868 900 

9.868 785 
.868 670 
.868 555 
.868 440 
.868 324 

9.868 209 
.868 093 
.867 978 
.867 862 
•867 747 

9.867 631 
.867515 
•867 399 
.867 283 
.867 167 

9.867051 
.866 935 
.866819 
.866 703 
.866 586 

9.866 470 
.866 353 
.866 237 
.866 120 
.866 004 

9.865 887 
.865 770 
.865 653 
.865 536 
.865419 

9.865 302 
.865 185 
.865 068 
.864 950 
.864 833 

9.864716 
.864 598 
.864481 
.864 363 
.864 245 

9.864 127 
Sin. 



D.l' 



D. 1". 
~47° 



Tan. 



9-954 437 
•954 691 
.954 946 
.955 200 
•955 454 

9-955 7°8 
•955 96i 
.956215 
.956 469 
•956 723 

9-95 6 977 
•957 231 
•957 485 
•957 739 
•957 993 

9.958 247 
.958 500 

•958 754 
.959008 
.959 262 

9-959 5 l6 
•959 7 6 9 
.960023 
.960 277 
.960 530 

9.960 784 
.961 038 
.961 292 

•961 545 
.961 799 

9.962052 
.962 306 
.962 560 
.962813 
.963 067 

9.963 320 

•963 574 
.963 828 
.964081 
•964 335 

9.964 588 
.964 842 
•965 095 
•965 349 
.965 602 

9-965 855 
.966 109 
.966 362 
.966616 
.966 869 

9.967 123 

.967 376 
.967 629 
.967 883 
.968 136 

9.968 389 
.968 643 
.968 896 

•969 149 
.969 403 

9.969 656 
Cot 



D. 1". 



4-23 
4-25 
4-23 
4-23 
4-23 
4.22 
4-23 
4-23 
4-23 
4-23 
4-23 
4-23 
4-23 
4-23 
4-23 
4.22 
4-23 
4-23 
4-23 
4-23 
4.22 
4-23 
4-23 
4.22 

4-23 
4-23 
4-23 
4.22 

4-23 
4.22 

4-23 
4-23 
4.22 

4-23 
4.22 

4-23 
4-23 
4.22 

4-23 
4.22 

4-23 
4.22 

4.23 
4.22 
4.22 

4-23 
4.22 

4-23 
4.22 

4-23 
4.22 
4.22 

4-23 
4.22 
4.22 

4-23 
4.22 
4.22 

4.23 
4.22 

D. 1". 



Cot. 



0.045 563 


60 


.045 309 


59 


•045 °54 


58 


.044 800 


57 


.044 546 


56 


0.044 292 


55 


.044 039 


54 


.043 785 


53 


.043 53i 


52 


.043 277 


5i 


0.043 023 


5o 


.042 769 


49 


.042515 


48 


.042 261 


47 


.042 007 


46 


0.041 753 


45 


.041 500 


44 


.041 246 


43 


.040 992 


42 


.040 738 


4i 


0.040 484 


40 


.040231 


39 


•039 977 


38 


•039 7 2 3 


37 


.039 470 


36 


0.039 216 


35 


.038 962 


34 


.038 708 


33 


•038 455 


32 


.038 201 


3i 


0.037 948 


30 


.037 694 


29 


.037 440 


28 


.037 187 


27 


•036 933 


26 


0.036 680 


25 


.036 426 


24 


.036172 


23 


.035 919 


22 


.035 665 


21 


0.035 412 


20 


•035 J 58 


19 


•034 905 


18 


.034651 


17 


•034 398 


16 


o-034 145 


15 


•033 891 


14 


.033 638 


13 


.033 384 


12 


.033 131 


11 


0.032 877 


10 


.032 624 


9 


.032371 


8 


.032117 


7 


.031 864 


6 


0.031 611 


5 


.031 357 


4 


.031 104 


3 


.030851 


2 


.030 597 


1 


0.030 344 






Tan. 



LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 223 

43° 



o 

i 

2 

3 
4 

5 
6 

7 
8 

9 

10 

ii 

12 

13 
14 

15 
16 

17 
18 

19 

20 
21 
22 
23 

24 

25 

26 

27 

28 

29 

30 
31 
32 

33 
34 
35 
36 
37 
38 
39 
40 
4i 
42 
43 
44 

45 
46 

47 
48 

49 
5o 
5i 
52 
53 
54 

55 
56 
57 
58 
59 
60 



Sin. 



9- 



9 



833 783 

833 919 
834054 
834189 

834 325 
834 460 
834 595 
834 73o 
834 865 

834 999 

835 134 
835 26 9 
835 403 
835 538 
835 672 
835 807 

835 94i 
836075 

836 209 
836 343 
836477 
836 61 1 

836 745 
836878 
837012 

837 146 
837 279 
837412 

837 546 
837 6 79 
837812 

837 945 
838078 

838 21 1 

838 344 

838 477 
838610 

838 742 
838875 

839 007 

839 140 

839 272 
839 404 
839 536 
839 668 

839 800 

839 932 

840 064 
840 196 
840 328 

840 459 
840 591 
840 722 
840 854 

840 985 

841 116 
841 247 
841 378 
841 509 
841 640 
841 771 



D, 1". 



2.27 
2.25 
2.25 
2.27 
2.25 
2.25 
2.25 
2.25 
2.23 
2.25 
2.25 
2.23 
2.25 
2.23 
2.25 
2.23 
2.23 
2.23 
2.23 
2.23 
2.23 
2.23 
2.22 
2.23 
2.23 
2.22 
2.22 
2.23 
2.22 
2.22 
2.22 
2.22 
2.22 
2.22 
2.22 
2.22 
2.20 
2.22 
2.20 
2.22 
2.20 
2.20 
2.20 
2.20 
2.20 
2.20 
2.20 
2.20 
2.20 
2.18 
2.20 
2.18 
2.20 
2.18 
2.18 
2.18 
2.18 
2.18 
2.18 
2.18 

D. 1", 



Cos. 



9.864 127 
.864010 
.863 892 
.863 774 
.863 656 

9.863 538 
.863 419 
.863 301 
.863 183 
.863 064 

9.862 946 
.862 827 
.862 709 
.862 590 
.862471 

9.862 353 
.862 234 
.862 115 
.861 996 
.861 877 

9.861 758 
.861 638 
.861 519 
.861 400 
.861 280 

9.861 161 
.861 041 
.860 922 
.860 802 
.860682 

9.860 562 
.860 442 
.860 322 
.860 202 
.860082 

9.859 962 
.859 842 
.859 721 
.859 601 
.859 480 

9.859 360 
.859 239 
.859119 
•858 998 
.858877 

9.858 756 
.858635 
.858514 
•858 393 
.858272 

9.858 151 
.858029 
.857 908 
.857 786 
.857 665 

9-857 543 
.857 422 
.857 300 
.857178 
.857056 

9-856 934 

Sin, 



D. 1". 



2.00 
1.98 
2.00 
2.00 
2.00 
2.00 
2.00 
2.00 
2.00 
2.00 
2.00 
2.02 
2.00 
2.02 
2.00 
2.02 
2.00 
2.02 
2.02 
2.02 
2.02 
2.02 
2.02 
2.02 
2.02 
2.03 
2.02 
2.03 
2.02 
2.03 
2.02 
2.03 
2.03 
2.03 
2.03 

D, I", 

~46° 



Tan. 



9.969 656 
.969 909 
.970 162 
.970416 
.970 669 

9.970922 

•971 175 
.971 429 
.971 682 
•97i 935 

9.972188* 
.972 441 
.972 695 
.972 948 
•973 201 

9-973 454 
•973 7°7 
.973 960 

.974 213 

•974 466 

9.974 720 

•974 973 
.975 226 

•975 479 
•975 732 

9-975 985 
.976 238 
.976491 

•976 744 
.976997 

9.977 250 
•977 503 
•977 756 
.978 009 
.978 262 

9-978 5J5 
.978 768 
.979021 
•979 274 
•979 527 

9.979 780 
.980 033 
.980 286 
.980 538 
.980 791 

9.981 044 
.981 297 
.981 550 
.981 803 
.982 056 

9.982 309 
.982 562 
.982814 
.983 067 
•983 320 

9-983 573 
.983 826 
.984079 

•984 332 
.984 584 

9-984 837 

Cot. 



D. 1". 



4.22 
4.22 

4-23 
4.22 

4.22 
4.22 

4-23 
4.22 
4.22 
4.22 
4.22 

4-23 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4-23 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.20 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.20 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.20 
4.22 

D. 1". 



Cot. 



0.030 344 


60 


.030091 


59 


.029 838 


58 


.029 584 


57 


.029 331 


5b 


0.029 078 


55 


.028 825 


54 


.028 571 


53 


.028318 


52 


.028 065 


5i 


0.027 812 


50 


•027 559 


49 


.027 305 


48 


.027 052 


47 


.026 799 


46 


0.026 546 


45 


.026 293 


44 


.026 040 


43 


.025 787 


42 


•025 534 


4i 


0.025 2 8° 


40 


.025 027 


39 


•024 774 


38 


.024 521 


37 


.024 268 


36 


0.024015 


35 


.023 762 


34 


.023 509 


33 


.023 256 


32 


•023 003 


3i 


0.022 750 


30 


.022 497 


29 


.022 244 


28 


.021 991 


27 


.021 738 


26 


0.021 485 


25 


.021 232 


24 


.020 979 


23 


.020 726 


22 


.020 473 


21 


0.020 220 


20 


.019 967 


19 


.019714 


18 


.019 462 


17 


.019 209 


16 


0.018 956 


15 


.018 703 


14 


.018450 


13 


.018 197 


12 


.017944 


11 


0.017 691 


10 


.017438 


9 


.017 186 


8 


.016933 


7 


.016680 


6 


0.016427 


5 


.016 174 


4 


.015 921 


3 


.015668 


2 


.015 416 


1 


0.015 163 






Tan. 



M. 



224 LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 

44° 



o 

i 

2 

3 
4 

5 
6 

7 
8 

9 
io 

ii 

12 

13 

14 

15 
16 

17 
18 

19 

20 

21 
22 
23 

24 

25 
26 

27 
28 

29 

30 
31 
32 

33 
34 

35 
36 
37 
38 
39 
40 
4i 
42 
43 
44 

45 
46 

47 
48 

49 
50 
5i 
52 
53 
54 
55 
56 
57 
58 
59 
60 



Sin. 



841 771 

841 902 

842 033 
842 163 
842 294 
842 424 

842 555 
842 685 
842815 

842 946 

843 076 
843 206 

843 336 
843 466 

843 595 
843 7 2 5 
843855 

843 984 

844 114 
844 243 
844372 
844 502 
844631 
844 760 

844 889 

845 018 
845 J 47 
845 276 
845 405 
845 533 
845 662 

845 79o 
845 919 
.846 047 
.846175 
9.846 304 
.846 432 
.846 560 
.846 688 
.846816 

9.846 944 
.847071 
.847 199 
.847 327 
•847 454 

9.847 582 
.847 709 
.847 836 
.847 964 
.848091 

9.848218 
.848 345 
.848 472 
.848 599 
.848 726 

9.848 852 
•848 979 
.849 106 
.849 232 
•849 359 

9.849 485 



D. 1". 



9- 



2.18 
2.18 
2.17 
2.18 
2.17 
2.18 
2.17 
2.17 
2.18 
2.17 
2.I7 
2.17 
2.17 

2.15 
2.I7 

2.I7 
2.15 
2.I7 
2.I5 
2.I5 
2.17 
2.I5 

2.15 
2.15 
2.I5 
2.15 

2.15 
2.I5 

2.13 
2.15 

2.13 
2.15 
2.13 
2.13 

2.15 



13 
13 
13 
13 
13 
12 
13 
13 
12 

13 

12 
12 

13 

2.12 
2.12 
2.12 
2.12 
2.12 
2.12 
2.IO 
2.12 
2.12 
2.IO 
2.12 
2.IO 



D, 1". 



Cos. 



9.856934 
.856812 
.856 690 
.856 568 
.856 446 

9.856 323 
.856 201 
.856078 
•855 956 
.855 ^33 

9-855 7ii 
.855 588 

.855 465 
.855 342 
.855 219 

9.855 096 

.854 973 
.854850 
.854 727 
.854 603 

9.854 480 
.854 35 6 
•854 233 
.854 109 
.853986 

9.853 862 

.853 738 
.853614 

•853 49o 
.853 366 

9.853 242 
.853118 
.852 994 
.852869 
.852 745 

9.852620 
.852496 

.852 37 1 
.852 247 
.852 122 

9.85I997 
.851 872 

.851 747 
.851 622 
.851 497 

9.851 372 
.851 246 
.851 121 
.850 996 
.850870 

9.850 745 
.850619 
.850493 
.850 368 
.850 242 

9.850 116 
.849 990 
.849 864 
.849 738 
.849 611 

9.849 485 
Sin. 



D. 1". 



2.03 
2.03 
2.03 
2.03 
2.05 
2.03 
2.05 
2.03 
2.05 
2.03 
2.05 
2.05 
2.05 
2.05 
2.05 
2.05 
2.05 
2.05 
2.07 
2.05 
2.07 
2.05 
2.07 
2.05 
2.07 
2.07 
2.07 
2.07 
2.07 
2.07 
2.07 
2.07 
2.08 
2.07 
2.08 
2.07 
2.08 
2.07 
2.08 
2.08 
2.08 
2.08 
2.08 
2.08 
2.08 
2.10 
2.08 
2.08 
2.10 
2.08 
2.10 
2.10 
2.08 
2.10 
2.10 
2.10 
2.10 
2.10 
2.12 
2.10 

D. 1". 

45^ 



Tan. 



9.984 837 
.985 090 
•985 343 
•985 596 
.985 848 

9.986 101 
.986 354 
.986 607 
.986 860 
.987 112 

9-987 365 
.987618 
.987 871 
.988 123 
.988376 

9.988 629 
.988 882 
.989 134 
.989 387 
.9S9 640 

9.989 893 
.990 145 
.990 398 
.990651 
.990 903 

9.991 156 
.991 409 
.991 662 
.991914 
.992 167 

9.992 420 
.992 672 
.992 925 
.993178 
•993 43 1 

9-993 683 
•993 936 
•994 189 
•994 44i 
•994 694 

9.994 947 
•995 *99 
•995 452 
•995 7°5 
•995 957 

9.996 210 
.996 463 
.996715 
.996 968 
.997 221 

9-997 473 
•997 726 
•997 979 
.998 231 
.998 484 

9-998 737 
.998 989 
.999 242 

•999 495 
•999 747 



Cot. 



D. 1". 



4.22 
4.22 
4.22 
4.20 
4.22 
4.22 
4.22 
4.22 
4.20 
4.22 
4.22 
4.22 
4.20 
4.22 
4.22 
4.22 
4.20 
4.22 
4.22 
4.22 
4.20 
4.22 
4.22 
4.20 
4.22 
4.22 
4.22 
4.20 
4.22 
4.22 
4.20 
4.22 
4.22 
4.22 
4.20 
4.22 
4.22 
4.20 
4.22 
4.22 
4.20 
4.22 
4.22 
4.20 
4.22 
4.22 
4.20 
4.22 
4.22 
4.20 
4.22 
4.22 
4.20 
4.22 
4.22 
4.20 
4.22 
4.22 
4.20 
4.22 

D. 1". 



Cot. 



0.015 l6 3 
.014 910 
.014657 
.014404 
.014 152 

0.013899 
.013 646 

•OI3 393 
.013 140 
.012888 

0.012 635 
.012 382 
.012 129 
.011 877 
.011 624 

0.011 371 
.011 118 
.010 866 
.010 613 
.010 360 

0.010 107 
.009 855 
.009 602 
.009 349 
.009 097 

0.008 844 
.008 591 
.008 338 
.008 086 
.007 833 

0.007 58° 
.007 328 
.007 075 
.006 822 
.006 569 

0.006 317 
.006 064 
.005 811 

.005 559 
.005 306 

0.005 °53 
.004 801 
.004 548 
.004 295 
.004 043 

0.003 790 

•003 537 
.003 285 
.003 032 
.002 779 

0.002 527 
.002 274 
.002 021 
.001 769 
.001 516 

0.00 1 263 
.001 on 
.000 758 
.000 505 
.000 253 

0.000 000 
Tan. 



TABLE XIX 

NATURAL SINES, COSINES, TANGENTS, 
AND COTANGENTS, 

FOR EVERY 

DEGREE AND MINUTE FROM 0° TO 90*. 



226 NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 

0° 1° 2° 



Sin. 



Cos, 



Tan. 



Oot. 



Sin. Cos, Tan, Cot. 



Sin. Cos. Tan. 



Cot, 



o 

i 

2 

3 
4 

5 
6 

7 
8 

9 
io 
ii 

12 

13 
14 

15 
16 

17 
18 

19 

20 
21 
22 

23 ! 

24 

25 
26 

27 
28 

29 

30 
3i 

32 

33 
34 

35 
36 
37 
38 
39 
40 
41 
42 
43 
44 

45 
46 

47 
48 

49 
50 
5i 
52 
53 
54 

55 
56 
57 
58 
59 
60 



029 
058 
087 
116 
.00145 

175 

204 

233 

262 



1. 0000 
000 
000 
000 
000 

1 .0000 
000 
000 
000 
000 



00000 
029 

058 

087 

116 

.00145 

175 
204 

233 
262 



00291 


r.0000 . 


00291 


320 


99999 


320 


349 


999 


349 


37* 


999 


37* 


407 


999 


407 


00436 


99999 . 


00436 


465 


999 


405 


495 


999 


495 


5 2 4 


999 


524 


553 


998 


553 


00582 


99998 .00582 


611 


998 


611 


640 


998 


640 


669 


998 


669 


698 


998 


698 



.00727 

756 

814 
844 

.00873 
902 

93i 
960 

989 
.01018 
047 
076 
105 
i34 
.01164 

i93 
222 

251 
280 

.01309 
338 
367 
396 
425 

.01454 
483 
5i3 
542 
57i 

.01600 
629 
658 
687 
716 

•01745 



,99997 
997 
997 
997 
996 

.99996 
996 
996 
995 
995 

•99995 
995 
994 
994 
994 

•99993 
993 
993 
992 

992 
.99991 
991 
991 
990 
990 
,99989 
989 
989 



•99987 
987 
986 
986 
985 

.99985 



00 

3437-7 
1718.9 
1 H5-9 
859-44 

687.55 
572.96 
491. 11 
429.72 
381.97 

343-77 
312.52 
286.48 
264.44 
245-55 
229.18 
214.86 
202.22 
190.98 
180.93 

171.89 
163.70 
156.26 
149.47 
143.24 

I37-5I 

132.22 
127.32 
122.77 
118.54 

H4-59 
110.89 

10743 
104.17 

IOI.II 

.01018 98.218 

047 95-489 
076 92.908 
105 90.463 
135 88.144 
.01164 85.940 
193 83.844 
222 81.847 

251 79-943 
280 78.126 

.01309 76.390 
338 74.729 
367 73-139 
396 71.615 

425 7°- I 53 

.01455 68.750 

484 67.402 

51366.105 

542 64.858 

57i 63.657 

.01600 62.499 

629 61.383 

658 60.306 

687 59.266 

716 58.261 

.01746 57.290 



.00727 
756 
785 
815 

844 

.00873 

902 

93i 
960 

989 



.01745 
774 
803 

832 
862 

.01891 
920 
949 
978 

.02007 

.02036 
065 

094 
123 

152 
.02181 
211 
240 
269 
298 
•02327 
356 
385 
414 

443 
.02472 

5°i 
53o 
560 

589 

.02618 
647 
676 
705 
734 

•02763 
792 
821 
850 
879 

.02908 
938 
967 
996 

.03025 

•03054 
083 
112 
141 
170 

.03199 
228 

257 
286 
3i6 

•03345 
374 
403 
432 
461 

.03490 



984 
984 
983 
983 

.99982 
982 
981 
980 
980 

.99979 
979 
978 
977 
977 

.99976 
976 
975 
974 
974 

•99973 
972 
972 
971 
97° 

•99969 
969 
968 

967 
966 

.99966 
965 
964 
963 
963 

.99962 
961 
960 
959 
959 

.99958 
957 
956 
955 
954 

•99953 
952 
952 
95 1 
95° 

.99949 
948 

947 
946 

945 
•99944 
943 
942 
941 
940 

•99939 



.01746 57.290 

775 56-35 1 
804 55.442 

^33 54-56i 
862 53.709 

.01891 52.882 
920 .081 

949 5 J -303 
978 50.549 
.02007 49.816 
.02036 49.104 
066 48.412 
095 47-740 
124 .085 

153 46.449 
,02182 45.829 
211 .226 
240 44.639 
269 .066 
298 43-5°8 
.02328 42.964 

357 -433 
386 41.916 
415 .411 
44440.917 

.02473 40.436 
502 39.965 
531 .506 
560 .057 
589 38.618 

.02619 38.188 
648 37.769 
677 .358 

. 706 36.956 
735 . -563 

.02764 36.178 
793 35-8oi 
822 .431 
851 .070 
881 34.715 

.02910 34.368 
939 -027 
968 33.694 
997 -366 

.03026 .045 

.03055 32.730 
084 .421 
114 .118 
143 31.821 
172 .528 



,03201 
230 

259 
288 

317 
03346 
376 
405 
434 
463 
.03492 



31.242 

30.960 

.683 

.412 

•145 

29.882 

.624 

•37i 
.122 

28.877 
28.636 



.03490 
5*9 
548 

577 
606 

•03635 
664 

693 

723 
752 

.03781 
810 

839 

868 

897 
.03926 

955 

984 
■04013 

042 

.04071 
100 
129 

i59 
188 

.04217 
246 
275 
304 
333 

.04362 

39i 
420 

449 
478 

.04507 
536 
565 
594 
623 

•04653 
682 
711 
740 
769 

.04798 
827 
856 
885 
914 

.04943 
972 

.05001 
030 
059 

.05088 , 
117 
146 

175 
205 

.05234 , 



99939 -03492 28.636 
938 521 -399 
937 55° - l6 6 
936 579 27.937 
935 609 .712 

■99934 -03638 27.490 
933 667 .271 
932 696 .057 
931 725 26.845 
930 754 .637 

.99929 .03783 26.432 
927 812 .230 
926 842 .031 
925 871 25.835 
924 900 .642 

.99923 .03929 25.452 
922 958 .264 
921 987 .080 
919 .04016 24.898 
918 046 .719 



99917 
916 
915 
913 

912 

9991 1 
910 
909 
907 
906 

99905 
904 
902 
901 
900 



897 
896 
894 
893 



890 



886 

.99885 
883 
882 
881 
879 

.99878 
876 

875 
873 
872 

,99870 
869 
867 
866 
864 

99863 



.04075 
104 

133 
162 
191 
.04220 
250 

279 
308 

337 
.04366 

395 
424 

454 
483 
.04512 
54i 
57o 
599 
628 

.04658 
687 
716 
745 
774 

.04803 

833 
862 
891 
920 
.04949 

978 
.05007 

037 

066 

.05095 
124 

153 
182 
212 



24.542 
.368 
.196 
.026 

23.859 

23.695 
•532 
.372 
.214 
.058 

22.904 

•752 
.602 

•454 
.308 

22.164 

.022 

21.881 

•743 
.606 

21.470 

•337 
.205 

•075 
20.946 

20.819 
.693 
•569 
.446 

•325 
20.206 

.087 
19.970 

.855 

.740 
19.627 

.516 

.405 
.296 
.188 



,05241 19.081 



Cos. Sin. Cot, Tan. 

89° ~ 



Cos. Sin. Cot. 

88° 



Tan. 



Cos. 



Sin. Cot. Tan. 

87° 



NATURAL SINES, COSINES, TANGENTS, 
3° 4° 



AND COTANGENTS. 
5° 



227 



M. 
o 


Sin. Cos. Tan. Cot. 


Sin. Cos. Tan. Cot. 


Sin. Cos. Tan. Cot. 




.05234 .99863 .05241 19.081 


.06976 .99756 .06993 I4-30I 


.08716 .99619 .08749 n.430 


60 


I 


263 861 270 18.976 


.07005 754 .07022 .241 


745 6l 7 778 .392 


59 


2 


292 860 299 .871 


034 75 2 051 .182 


774 614 807 .354 


58 


3 


321 858 328 .768 


063 750 080 .124 


803 612 837 ,316 


57 


4 


350 857 357 .666 


092 748 no .065 


831 609 8b6 .279 


56 


5 


•05379 -99855 -05387 18.564 


.07121 .99746 .07139 14.008 


.08860 .99607 .08895 11-242 


55 


6 


408 854 416 .464 


150 744 168 13.951 


889 604 925 .205 


54 


7 


437 852 445 .366 


179 742 197 .894 


918 602 954 .168 


53 


8 


466 851 474 .268 


208 740 227 .838 


947 599 983 -132 


52 


9 


495 849 503 .171 


237 738 256 .782 


976 596 .09013 .095 


5i 


IO 


•05524 .99847 .05533 18.075 


.07266 .99736 .07285 13.727 


.09005 .99594 .09042 n.059 


50 


ii 


553 846 562 17.980 


295 734 3H .672 


034 59i 071 .024 


49 


12 


582 844 591 .886 


324 731 344 .617 


063 588 101 10.988 


48 


13 


611 842 620 .793 


353 729 373 .563 


092 586 130 .953 


47 


14 


640 841 649 .702 


382 727 402 .510 


121 583 159 .918 


46 


15 


.05669 .99839 .05678 1 7.61 1 


.07411 .99725 .07431 13.457 


.09150 .99580 .09189 10.883 


45 


16 


698 838 708 .521 


440 723 461 .404 


179 578 218 .848 


44 


17 


727 836 737 .431 


469 721 490 .352 


208 575 247 .814 


43 


18 


756 834 766 .343 


498 719 519 .300 


237 572 277 .780 


42 


19 


785 ^33 795 -256 


527 716 548 .248 


266 570 306 .746 


4i 


20 


.05814 .99831 .05824 17.169 


•07556 .997H -07578 I3-I97 


•09295 -995 6 7 .09335 IO -7 12 


40 


21 


844 829 854 .084 


585 712 607 .146 


324 564 365 .678 


39 


22 


873 827 883 16.999 


614 710 636 .096 


353 5 62 394 -645 


38 


23 


902 826 912 .915 


643 708 665 .046 


382 559 423 .612 


37 


24 


931 824 941 .832 


672 705 695 12.996 


411 55 6 453 -579 


36 


25 


.05960 .99822 .05970 16.750 


.07701 .99703 .07724 12.947 


.09440 .99553 .09482 10.546 


35 


26 


989 821 999 .668 


73o 7 QI 753 -898 


469 551 511 .514 


34 


27 


.06018 819 .06029 -587 


759- 699 782 .850 


498 548 541 .481 


33 


28 


047 817 058 .507 


788 696 812 .801 


527 545 570 .449 


32 


29 


076 815 087 .428 


817 694 841 .754 


^ 55 6 542 600 .417 


3i 


30 


.06105 -99813 .06116 16.350 


.07846 .99692 .07870 12.706 


•09585 -9954Q .09629 10.385 


30 


31 


134 812 145 .272 


875 689 899 .659 


614 537 658 .354 


29 


32 


163 810 175 .195 


904 687 929 .612 


642 534 688 .322 


28 


33 


192 808 204 .119 


933 685 958 .566 


671 53i 7 l 7 -291 


27 


34 


221 806 233 .043 


962 683 987 .520 


700 528 746 .260 


26 


35 


.06250 .99804 .06262 15.969 


.07991 .99680 .08017 12.474 


.09729 .99526 .09776 10.229 


25 


36 


279 803 291 .895 


.08020 678 046 .429 


758 5 2 3 805 .199 


24 


37 


308 801 321 .821 


049 676 075 .384 


787 520 834 .168 


23 


33 


337 799 35° -748 


078 673 104 .339 


816 517 864 .138 


22 


39 


366 797 379 .676 


107 671 134 .295 


845 5H 893 .108 


21 


40 


•06395 -99795 .06408 15.605 


.08136 .99668 .08163 12.251 


.09874 .99511 .09923 10.078 


20 


4i 


424 793 438 .534 


165 666 192 .207 


903 5° 8 952 .048 


19 


42 


453 792 467 464 


194 664 221 .163 


932 506 981 .019 


18 


43 


482 .790 496 .394 


223 661 251 .120 


961 503 .10011 9.9893 


17 


44 


511 788 525 .325 


252 659 280 .077 


990 500 040 .9601 


16 


45 


-06540 .99786..06554 15.257 


.08281 .99657 .08309 12.035 


.10019 .99497 .10069 9-93 10 


15 


46 


569 784 584 .189 


310 654 339 n.992 


048 494 099 .9021 


14 


47 


598 782 613 .122 


339 652 368 .950 


077 491 128 .8734 


13 


48 


627 780 642 .056 


368 649 397 .909 


106 488 158 .8448 


12 


49 


656 778 671 14.990 


397 647 427 .867 


135 485 187 .8164 


n 


5o 


.06685 -99776 .06700 14.924 


.08426 .99644 .08456 11.826 


.10164 .99482 .10216 9.7882 


10 


5i 


714 774 730 .860 


455 642 485 .785 


192 479 246 .7601 


9 


52 


743 772 759 -795 


484 639 514 .745 


221 476 275 .7322 


8 


53 


773 770 788 .732 


5 r 3 637 544 -7°5 


250 473 3°5 -7044 


7 


54 


802 768 817 .669 


542 635 573 .664 


279 47° 334 -6768 


6 


55 


.06831 .99766 .06847 14-606 


.08571 .99632 .08602 11.625 


.10308 .99467 .10363 9.6493 


5 


56 


860 764 876 .544 


600 630 632 .585 


337 464 393 -6220 


4 


57 


889 762 905 .482 


629 627 661 .546 


366 461 422 -5949 


3 


58 


918 760 934 .421 


658 625 690 .507 


395 458 452 .5679 


2 


59 


947 758 963 .361 


687 622 720 .468 


424 455 481 -54" 


X 


60 


•06976 .9975 6 -06993 H-30I 


.08716 .99619 .08749 n.430 


.10453 .99452 .10510 9.5144 




M. 


, 


Cos. Sin. Cot. Tan, 


Cos. Sin. Cot, Tan. 


Cos. Sin. Cot. Tan. 



86° 



85° 



84 c 



228 NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
6° 7° 8° 



M. 


Sin. Cos. Tan. Cot. 


Sin. Cos. Tan. Cot. 


Sin. Cos. Tan. Cot. 




o 


•10453 -9945 2 -105109.5144 


.12187 -99255 - I2 278 8.1443 


.13917 .99027 .14054 7.1154 


60 


i 


482 449 540 .4878 


216 251 308 .1248 


946 023 084 .1004 


59 


2 


511 446 569 .4614 


245 248 338 .1054 


975 019 113 .0855 


58 


3 


540 443 599 .4352 


274 244 367 .0860 


.14004 015 143 .0706 


57 


4 


569 440 628 .4090 


302 240 397 .0667 


033 on 173 .0558 


56 


5 


•io597 -99437 - Io6 57 9-3831 


.12331 .99237 .12426 8.0476 


.14061 .99006 .14202 7.0410 


55 


6 


626 434 687 .3572 


360 233 456 .0285 


090 002 232 .0264 


54 


7 


655 43i 7 l6 -3315 


389 230 485 .0095 


119 .98998 262 .0117 


53 


8 


684 428 746 .3060 


418 226 515 7.9906 


148 994 291 6.9972 


52 


9 


7 l 3 424 775 - 28 ° 6 


447 222 544 .9718 


177 990 321 .9827 


5i 


10 


.10742 .99421 .10805 9.2553 


.12476 .99219 .12574 7.9530 


.14205 .98986 .14351 6.9682 


5o 


ii 


771 418 834 .2302 


504 215 603 .9344 


234 982 381 .9538 


49 


12 


800 415 863 .2052 


533 211 633 .9158 


263 978 410 .9395 


48 


13 


829 412 893 .1803 


562 208 662 .8973 


292 973 440 .9252 


47 


14 


858 409 922 .1555 


591 204 692 .8789 


320 969 470 .9110 


46 


15 


.10887 -99406 .10952 9.1309 


.12620 .99200 .12722 7.8606 


.14349 .98965 .14499 6.8969 


45 


16 


916 402 981 .1065 


649 197 75 1 -8424 


378 961 529 .8828 


44 


17 


945 399 .1101 1 .0821 


678 193 781 .8243 


407 957 559 -8687 


43 


18 


973 396 040 .0579 


706 189 810 .8062 


436 953 588 .8548 


42 


19 


.11002 393 070 .0338 


735 186 840 .7882 


464 948 618 .8408 


4i 


20 


.11031 .99390 .11099 9.0098 


.12764 .99182 .12869 7-77°4 


.14493 -98944 .14648 6.8269 


40 


21 


060 386 128 8.9860 


793 178 899 .7525 


522 940 678 .8131 


39 


22 


089 383 158 -9623 


822 175 929 .7348 


55 1 936 7°7 -7994 


38 


23 


118 380 187 .9387 


851 171 958 .717 1 


580 931 737 .785 6 


37 


24 


147 377 217 .9152 


880 167 988 .6996 


608 927 767 .7720 


36 


25 


.11176 .99374 .11246 8.8919 


.12908 .99163 .13017 7.6821 


.14637 .98923 .14796 6.7584 


35 


26 


2 °5 37° 2 76 -8686 


937 160 047 .6647 


666 919 826 .7448 


34 


27 


234 367 305 -8455 


966 156 076 .6473 


695 914 856 .7313 


33 


28 


263 364 335 -8225 


995 l 5 2 Io6 - 6 3°i 


723 910 886 .7179 


32 


29 


291 360 364 .7996 


.13024 148 136 .6129 


752 906 915 .7045 


3i 


30 


.11320 .99357 .113948.7769 


.13053 -99144 • 13*65 7-5958 


.14781 .98902 .14945 6.6912 


30 


31 


349 354 423 -7542 


081 141 195 .5787 


810 897 975 .6779 


29 


32 


378 351 452 -73I7 


no 137 224 .5618 


838 893 .15005 .6646 


28 


33 


407 347 482 .7093 


139 133- 254 .5449 


867 889 034 .6514 


27 


34 


436 344 511 .6870 


168 129 284 .5281 


896 884 064 .6383 


26 


35 


.11465 .99341 .11541 8.6648 


.13197 .99125 .13313 7.5113 


.14925 .98880 .15094 6.6252 


25 


36 


494 337 57° -6427 


226 122 343 .4947 


954 876 124 .6122 


24 


37 


5 2 3 334 600 .6208 


254 118 372 .4781 


982 871 153 .5992 


23 


38 


552 331 629 .5989 


283 114 402 .4615 


.15011 867 183 .5863 


22 


39 


580 327 659 .5772 


312 no 432 .4451 


040 863 213 .5734 


21 


40 


.11609 .99324.116888.5555 


.13341 .99106 .13461 7.4287 


.15069 .98858 .15243 6.5606 


20 


4i 


638 320 718 .5340 


370 102 491 .4124 


097 854 272 .5478 


19 


42 


667 3 J 7 747 -5 I2 6 


399 098 521 .3962 


126 849 302 .5350 


18 


43 


696 314 777 .4913 


427 094 550 .3800 


155 845 332 .5223 


17 


44 


725 310 806 .4701 


456 091 580 .3639 


184 841 362 .5097 


16 


45 


•"754 -993°7 .118368.4490 


.13485 .99087 .13609 7.3479 


.15212 .98836 .15391 6.4971 


15 


46 


783 303 865 .4280 


514 083 639 .3319 


241 832 421 .4846 


H 


47 


812 300 895 .4071 


543 °79 669 .3160 


270 82y 451 .4721 


13 


48 


840 297 924 .3863 


572 075 698 .3002 


299 823 481 .4596 


12 


49 


869 293 954 .3656 


600 071 728 .2844 


327 818 511 .4472 


11 


5o 


.11898 .99290 .11983 8.3450 


.13629 .99067 .13758 7.2687 


-I535 6 -98814 .15540 6.4348 


10 


5i 


927 286 .12013 .3245 


658 063 787 .2531 


385 809 570 .4225 


9 


52 


956 283 042 .3041 


687 059 817 .2375 


414 805 600 .4103 


8 


53 


985 279 072 .2838 


716 055 846 .2220 


442 800 630 .3980 


7 


54 


.12014 276 101 .2636 


744 051 876 .2066 


471 796 660 .3859 


6 


55 


.12043 .99272 .12131 8.2434 


•13773 -99047 -13906 7.1912 


.15500 .98791 .15689 6.3737 


5 


56 


071 269 160 .2234 


802 043 935 .1759 


529 787 7 : 9 -3617 


4 


57 


100 265 190 .2035 


831 039 965 .1607 


557 782 749 .3496 


3 


58 


129 262 219 .1837 


860 035 995 .1455 


586 778 779 .3376 


2 


59 


158 258 249 .1640 


889 031 .14024 .1304 


615 773 809 .3257 


1 


60 


.12187 .99255 -12278 8.1443 


.13917 .99027 .14054 7.1154 


.15643 .98769 .15838 6.3138 





Cos. Sin. Cot. Tan. 


Cos. Sin. Cot. Tan. 


Cos. Sin. Cot. Tan. 


M. 



83° 



82° 



81° 



NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 229 
9° 10° 11° 



M. 


Sin. Cos. Tan. Cot. 


Sin. Cos. Tan. Cot. 


Sin. Cos. Tan. Cot. 




o 


.15643 .98769 .15838 6.3138 


.17365 .98481 .176335.6713 


.19081 .98163 .19438 5.1446 


60 


i 


672 764 868 .3019 


393 476 663 .6617 


109 157 468 .1366 


59 


2 


701 760 898 .2901 


422 471 693 .6521 


138 152 498 .1286 


58 


3 


730 755 928 .2783 


451 466 723 .6425 


167 146 529 .1207 


57 


4 


75 8 75 1 95 8 - 2666 


479 461 753 -63 2 9 


195 Ho 559 .1128 


56 


5 


.15787 .98746 .15988 6.2549 


.17508 .98455 .17783 5.6234 


.19224.98135 .19589 5.1049 


55 


6 


816 741 .16017 .2432 


537 45° 813 .6140 


252 129 619 .0970 


54 


7 


845 737 °47 - 2 3!6 


565 445 843 .6045 


281 124 649 .0892 


53 


8 


873 73 2 °77 - 2200 


594 440 873 .5951 


309 118 680 .0814 


52 


9 


902 728 107 .2085 


623 435 903 -5857 


338 112 710 .0736 


5i 


IO 


.15931 -98723 -16137 6 - I 97° 


.17651 .9843 .i7933 5.5764 


.19366 .98107 .19740 5.0658 


50 


ii 


959 7 l8 l6 7 -1856 


680 425 963 .5671 


395 IGI 77° -0581 


49 


12 


988 714 196 .1742 


708 420 993 .5578 


423 096 801 .0504 


48 


13 


.16017 709 226 .1628 


737 414.18023 .5485 


452 090 831 .0427 


47 


14 


046 704 256 .1515 


766 409 053 .5393 


481 084 861 .0350 


46 


15 


.16074 .98700 .16286 6.1402 


.17794 .98404 .18083 5.5301 


.19509 .98079 .19891 5.0273 


45 


16 


103 695 316 .1290 


823 399 113 -5 2 °9 


538 073 921 .0197 


44 


17 


132 690 346 .1178 


852 394 143 .5118 


566 067 952 .0121 


43 


18 


160 686 376 .1066 


880 389 173 .5026 


595 061 982 .0045 


42 


19 


189 681 405 .0955 


909 383 203 .4936 


623 056.200124.9969 


4i 


20 


.16218 .98676 .16435 6.0844 


•17937 -98378 .18233 54845 


.19652 .98050 .20042 4.9894 


40 


21 


246 671 465 .0734 


966 373 263 .4755 


680 044 073 .9819 


39 


22 


275 667 495 .0624 


995 368 293 .4665 


709 039 103 .9744 


38 


23 


304 662 525 .0514 


.18023 362 323 .4575 


737 °33 i33 -9669 


37 


24 


333 657 555 .0405 


05 2 357 353 4486 


766 027 164 .9594 


36 


25 


.16361 .98652 .16585 6.0296 


.18081 .98352 .18384 5.4397 


.19794 .98021 .20194 4.9520 


35 


26 


390 648 615 .0188 


109 347 4H .4308 


823 016 224 .9446 


34 


27 


419 643 645 .0080 


138 341 444 .4219 


851 010 254 .9372 


33 


28 


447 6 38 674 5.9972 


166 336 474 .4131 


880 004 285 .9298 


32 


29 


476 633 704 .9865 


195 33i 5°4 4043 


908.97998 315 .9225 


3i 


30 


.16505 .98629 .16734 5.9758 


.18224 .98325 .18534 5.3955 


•19937 -9799 2 - 2 0345 4-9I5 2 


30 


3i 


533 624 764 .9651 


252 320 564 .3868 


965 987 376 .9078 


29 


32 


562 619 794 .9545 


281 315 594 .3781 


994 981 406 .9006 


28 


33 


591 614 824 .9439 


309 310 624 .3694 


.20022 975 436 .8933 


27 


34 


620 609 854 .9333 


338 304 654 .3607 


051 969 466 .8860 


26 


35 


.16648 .98604 .16884 5.9228 


.18367 .98299 .18684 5.3521 


.20079 .97963 .20497 4.8788 


25 


36 


677 600 914 .9124 


395 2 94 7H -3435 


108 958 527 .8716 


24 


37 


7° 6 595 944 -9019 


424 288 745 -3349 


136 95 2 557 -8644 


23 


38 


734 590 974 -8915 


452 283 775 .3263 


165 946 588 .8573 


22 


39 


763 585.17004 .8811 


481 277 805 .3178 


J 93 94° 618 .8501 


21 


40 


.16792 .98580 .17033 5.8708 


.18509 .98272 .18835 5.3093 


.20222 .97934 .20648 4.8430 


20 


4i 


820 575 063 .8605 


538 267 865 .3008 


250 928 679 .8359 


19 


42 


849 570 093 .8502 


567 261 895 .2924 


279 922 709 .8288 


18 


43 


878 565 123 .8400 


595 2 5 6 9 2 5 .2839 


307 916 739 .8218 


17 


44 


906 561 153 .8298 


624 250 955 .2755 


336 910 770 .8147 


16 


45 


•16935 -98556. 17183 5-8i97 


.18652 .98245 .18986 5.2672 


.20364 .97905 .20800 4.8077 


15 


46 


964 551 213 .8095 


681 240 .19016 .2588 


393 899 830 .8007 


14 


47 


992 546 243 .7994 


710 234 046 .2505 


421 893 861 .7937 


13 


48 


.17021 541 273 .7894 


738 229 076 .2422 


450 887 891 .7867 


12 


49 


05° 536 303 -7794 


767 223 106 .2339 


478 881 921 .7798 


11 


50 


.17078 .98531 .17333 5.7694 


.18795 .98218.191365.2257 


.20507 .97875 - 2 Q95 2 4-77 2 9 


10 


5i 


107 526 363 .7594 


824 212 166 .2174 


535 869 982 .7659 


9 


52 


136 5 21 393 -7495 


852 207 197 .2092 


563 863.21013 .7591 


8 


53 


164 si6 423 .7396 


881 201 227 .2011 


592 857 043 .7522 


7 


54 


193 5 11 453 -7 2 97 


910 196 257 .1929 


620 851 073 .7453 


6 


55 


.17222.98506.17483 5.7199 


.18938 .98190 .192S7 5.1848 


.20649.97845 .211044.7385 


5 


56 


2 5° 5 QI 5 r 3 -7 101 


967 l8 5 3 l 7 - l 7 6 7 


677 839 134 -7317 


4 


57 


2 79 496 543 -7 00 4 


995 *79 347 - l6 S6 


706 833 164 .7249 


3 


58 


308 491 573 .6906 


.19024 174 378 .1606 


734 827 195 .7181 


2 


59 


336 486 603 .6809 


052 168 408 .1526 


763 821 225 .7114 


1 


60 


.17365 .98481 .17633 5.6713 


.19081 .98163 .19438 5.1446 


.20791 .97815 .21256 4.7046 







Cos. Sin. Cot. Tan. 


Cos. Sin. Cot. Tan. 


Cos. Sin. Cot. Tan. 



80 



79° 



78 ( 



230 NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
12° 13° 14° 



M. 


Sin. Cos. Tan. Cot. 


Sin. Cos. Tan, Cot, 


Sin, Cos. Tan. Cot. 




o 


.20791 .97815 .212564.7046 


•22495 -97437 -230874.3315 


.24192 .97030 .24933 4.0108 


60 


i 


820 809 286 .6979 


523 430 117 .3257 


220 023 964 .0058 


59 


2 


848 803 316 .6912 


552 424 148 .3200 


249 015 995 .0009 


58 


3 


877 797 347 -6845 


580 417 179 .3143 


277 008 .25026 3.9959 


57 


4 


905 79i 377 -6779 


608 411 209 .3086 


305 001 056 .9910 


56 


5 


•20933 -977 8 4 .21408 4.6712 


■22637 -97404 .23240 4.3029 


-24333 -96994 -25087 3.9861 


55 


6 


962 778 438 .6646 


665 398 271 .2972 


362 987 118 .9812 


54 


7 


990 772 469 .6580 


693 39i 3 QI -2916 


390 980 149 .9763 


53 


8 


.21019 766 499 .6514 


722 384 332 .2859 


418 973 180 .9714 


52 


9 


047 760 529 .6448 


75° 378 363 -2803 


446 966 211 .9665 


5i 


10 


.21076 .97754 .21560 4.6382 


.22778 .97371 .23393 4.2747 


.24474 .96959 .25242 3.9617 


50 


ii 


104 748 590 .6317 


807 365 424 .2691 


5°3 952 273 .9568 


49 


12 


132 742 621 .6252 


835 35 8 455 -2635 


53i 945 304 -9520 


48 


13 


161 735 651 .6187 


863 351 485 .2580 


559 937 335 -947* 


47 


14 


189 729 682 .6122 


892 345 516 .2524 


587 93o 366 .9423 


46 


15 


.21218 .97723 .21712 4.6057 


.22920 .97338 -23547 4-2468 


.24615 .96923 .25397 3.9375 


45 


16 


246 717 743 .5993 


948 33i 578 .2413 


644 916 428 .9327 


44 


17 


275 7" 773 -5928 


977 325 608 .2358 


672 909 459 .9279 


43 


18 


303 705 804 .5864 


.23005 318 639 .2303 


700 902 490 .9232 


42 


19 


331 698 834 .5800 


°33 3 11 670 .2248 


728 894 521 .9184 


4i 


20 


.21360 .97692 .21864 4.5736 


.23062 .97304 .237004.2193 


.24756 .96887 .25552 3.9136 


40 


21 


388 686 895 -5673 


090 298 731 .2139 


784 880 583 .9089 


39 


22 


417 680 925 .5609 


118 291 762 .2084 


813 873 614 .9042 


38 


23 


445 6 73 95 6 -5546 


146 284 793 .2030 


841 866 645 .8995 


37 


24 


474 667 986 .5483 


175 278 823 .1976 


869 858 676 .8947 


36 


25 


.21502 .97661 .22017 4.5420 


•23203 .97271 .238544.1922 


.24897 .96851 .25707 3.8900 


35 


25 


530 655 047 .5357 


231 264 885 .1868 


925 844 738 .8854 


34 


27 


559 648 078 .5294 


260 257 916 .1814 


954 ^37 769 -8807 


33 


23 


587 642 108 .5232 


288 251 946 .1760 


982 829 800 .8760 


32 


29 


616 636 139 .5169 


316 244 977 .1706 


.25010 822 831 .8714 


3i 


30 


.21644 -97630 .22169 4.5107 


•23345 -97237 .240084.1653 


.25038 .96815 .25862 3.8667 


30 


31 


672 623 200 .5045 


373 230 039 .1600 


066 807 893 .8621 


29 


32 


701 617 231 .4983 


401 223 069 .1547 


094 800 924 .8575 


28 


33 


729 611 261 .4922 


429 217- 100 .1493 


122 793 955 .8528 


27 


34 


758 604 292 .4860 


458 210 131 .1441 


151 786 986 .8482 


26 


35 


.21786.97598 .223224.4799 


.23486 .97203 .24162 4.138S 


.25179 .96778 .26017 3-8436 


25 


36 


814 592 353 -4737 


514 196 193 .1335 


207 771 048 .8391 


24 


37 


843 5 8 5 3*3 4676 


542 189 223 .1282 


235 764 079 .8345 


23 


38 


871 579 414 .4615 


571 182 254 .1230 


263 756 no .8299 


22 


39 


899 573 444 -4555 


599 176 285 .1178 


291 749 141 .8254 


21 


40 


.21928 .97566 .22475 4-4494 


.23627 .97169 .24316 4.1 126 


.25320 .96742 .26172 3.8208 


20 


4i 


956 560 505 .4434 


656 162 347 .1074 


348 734 203 .8163 


19 


42 


985 553 536 -4373 


684 155 377 - io 22 


376 727 235 .8118 


18 


43 


•22013 547 567 .4313 


712 148 408 .0970 


404 719 266 .8073 


17 


44 


041 541 597 .4253 


740 141 439 .0918 


432 712 297 .8028 


16 


45 


.22070 97534 .22628 4.4194 


.23769 .97134 .24470 4-0867 


-25460 .96705 .26328 3.7983 


15 


46 


098 528 658 .4134 


797 I2 7 5 QI -0815 


488 697 359 .7938 


14 


47 


126 521 689 .4075 


825 120 532 .0764 


516 690 390 .7893 


13 


48 


155 5*5 7 J 9 4015 


853 "3 562 .0713 


545 682 421 .7848 


12 


49 


183 508 750 .3956 


882 106 593 .0662 


573 6_75_ 452 .7804 


n 


50 


.22212 .97502 .22781 4.3897 


.23910 .97100 .24624 4.061 1 


.25601 .96667 .26483 3.7760 


10 


5i 


240 496 811 .3838 


938 093 655 .0560 


629 660 515 .7715 


9 


52 


268 489 842 .3779 


g66 086 686 .0509 


657 653 546 .7671 


8 


53 


297 483 872 .3721 


995 079 717 -°459 


685 645 577 .7627 


7 


54 


325 476 903 .3662 


.24023 072 747 .0408 


713 638 608 .7583 


6 


55 


■22353 -9747° -22934 4-3604 


.24051 .97065 .24778 4-0358 


.25741 .96630 .26639 3.7539 


5 


56 


382 463 964 .3546 


079 058 809 .0308 


769 623 670 .7495 


4 


57 


410 457 995 .3488 


108 051 840 .0257 


798 615 701 .7451 


3 


58 


438 450 .23026 .3430 


136 044 871 .0207 


826 608 733 .7408 


2 


59 


467 444 056 .3372 


164 037 902 .0158 


854 600 764 .7364 


1 


60 


•22495 -97437 -23087 4-33I5 


.24192 .97030 -24933 4.0108 


.25882 .96593 .26795 3.7321 







Cos. Sin. Cot. Tan. 


Cos. Sin. Cot. Tan. 


Cos. Sin. Cot. Tan. 


M. 



77c 



76 c 



75° 



NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
15° 16° 17° 



231 



M. 


Sin. Cos. Tan, Oot. 


Sin. Cos. Tan. Cot. 


Sin. Cos. Tan. Cot, 




o 


.25882 .96593 .26795 3.7321 


.27564 .96126 .28675 3.4874 


.29237 .95630 .30573 3.2709 


60 


i 


910 585 826 .7277 


592 118 706 .4836 


265 622 605 .2675 


59 


2 


938 578 857 .7234 


620 no 738 .4798 


293 613 637 .2641 


58 


3 


966 570 888 .7191 


648 102 769 .4760 


321 605 669 .2607 


57 


4 


994 562 920 .7148 


676 094 801 .4722 


348 596 700 .2573 


56 


5 


.26022 .96555 .26951 3.7105 


.27704 .96086 .28832 3.4684 


•29376 -95588 .30732 3-2539 


55 


6 


°5o 547 982 .7062 


731 078 864 .4646 


404 579 764 -2506 


54 


7 


079 540 .27013 .7019 


759 °7° 895 .4608 


432 571 796 .2472 


53 


8 


107 532 044 .6976 


787 062 927 .4570 


460 562 828 .2438 


52 


9 


135 5 2 4 °7 6 - 6 933 


815 °54 958 -4533 


487 554 860 .2405 


5i 


IO 


.26163 .96517 .27107 3.6891 


.27843 .96046 .28990 3.4495 


• 2 95 I 5 -95545 -30891 3-237 1 


50 


ii 


191 509 138 .6848 


871 037 .29021 .4458 


543 536 923 -2338 


49 


12 


219 502 169 .6806 


899 029 053 .4420 


57i 5 2 8 955 -2305 


48 


13 


247 494 201 .6764 


927 021 084 .4383 


599 5 X 9 987 -2272 


47 


14 


275 486 232 .6722 


955 013 116 .4346 


626 511 .31019 .2238 


46 


15 


.26303 .96479 .27263 3.6680 


.27983 .96005 .29147 3.4308 


•29654.95502 -3 10 S l 3-2205 


45 


16 


331 471 294 .6638 


.28011 .95997 179 .4271 


682 493 083 .2172 


44 


17 


359 463 326 .6596 


039 989 210 .4234 


710 485 115 .2139 


43 


18 


387 456 357 .6554 


067 981 242 .4197 


737 476 147 -2106 


42 


19 


415 448 388 .6512 


095 972 274 .4160 


765 467 178 .2073 


41 


20 


.26443 -96440 .27419 3-647° 


.28123 .95964 .29305 3.4124 


•29793 -95459 -3 I2 io 3.2041 


40 


21 


47 1 433 45 J -6429 


15° 95 6 337 4087 


821 450 242 .2008 


39 


22 


500 425 482 .6387 


178 948 368 .4050 


849 44i 274 .1975 


38 


23 


528 417 513 .6346 


206 940 400 .4014 


876 433 306 .1943 


37 


24 


556 410 545 .6305 


234 93i 432 -3977 


904 424 338 .1910 


36 


25 


.26584 .96402 .27576 3.6264 


.28262 .95923 .29463 3.3941 


•2993 2 -95415 -3137° 3- J 878 


35 


26 


612 394 607 .6222 


290 915 495 .3904 


960 407 402 .1845 


34 


27 


640 386 638 .6181 


318 907 526 .3868 


987 398 434 .1813 


33 


28 


668 379 670 .6140 


346 898 558 .3832 


.30015 389 466 .1780 


32 


29 


696 371 701 .6100 


374 890 590 .3796 


043 380 498 .1748 


3i 


30 


.26724 .96363 .27732 3.6059 


.28402 .95882 .29621 3.3759 


.30071 .95372 .315303.1716 


30 


3i 


752 355 7 6 4 .6018 


429 874 653 .3723 


098 363 562 .1684 


29 


32 


780 347 795 .5978 


457 865 685 .3687 


126 354 594 .1652 


28 


33 


808 340 826 .5937 


485 857 716 .3652 


154 345 626 .1620 


27 


34 


836 332 858 .5897 


513 849 748 .3616 


182 337 658 .1588 


26 


35 


.26864 .96324 .27889 3.5856 


.28541 .95841 .29780 3.3580 


.30209 .95328 .31690 3.1556 


25 


36 


892 316 921 .5816 


569 832 811 .3544 


237 319 722 .1524 


24 


37 


920 308 952 .5776 


597 824 843 .3509 


265 310 754 .1492 


23 


38 


948 301 983 .5736 


625 816 875 .3473 


292 301 786 .1460 


22 


39 


976 293 .28015 .5696 


652 807 906 .3438 


320 293 818 .1429 


21 


4P 


.27004 .96285 .28046 3.5656 


.28680 .95799 -29938 3-3402 


•30348 .95284.318503-1397 


20 


4i 


032 277 077 .5616 


708 791 970 .3367 


376 275 882 .1366 


19 


42 


060 269 109 .5576 


736 782 .30001 .3332 


403 266 914 .1334 


18 


43 


088 261 140 .5536 


764 774 033 .3297 


431 257 946 .1303 


17 


44 


116 253 172 .5497 


792 766 065 .3261 


459 248 978 .1271 


16 


45 


.27144 .96246 .28203 3.5457 


.28820 .95757 .30097 3.3226 


.30486 .95240 .32010 3.1240 


15 


46 


172 238 234 .5418 


847 749 128 .3191 


514 231 042 .1209 


14 


47 


200 230 266 .5379 


875 740 160 .3156 


542 222 074 .1178 


13 


48 


228 222 297 .5339 


903 732 192 -3 I2 2 


570 213 106 .1146 


12 


49 


256 214 329 .5300 


931 724 224 .3087 


597 204 139 .1115 


11 


50 


.27284 .96206 .28360 3.5261 


•28959 .95715 -30255 3.3052 


.30625 .95195 .32171 3.1084 


10 


51 


312 198 391 .5222 


987 707 287 .3017 


653 186 203 .1053 


9 


52 


340 190 423 .5183 


.29015 698 319 .2983 


680 177 235 .1022 


8 


53 


368 182 454 .5144 


042 690 351 .2948 


708 168 267 .0991 


7 


54 


396 174 486 .5105 


070 681 382 .2914 


736 159 299 .0961 


6 


55 


.27424 ,96166 .28517 3.5067 


.29098 .95673 -30414 3-2879 


•30763 -95 * 5° -32331 3-0930 


5 


56 


452 158 549 -5° 28 


126 664 446 .2845 


791 142 363 .0899 


4 


57 


480 150 580 .4989 


154 656 478 .2811 


819 133 396 .0868 


3 


58 


508 142 612 .4951 


182 647 509 .2777 


846 124 428 .0838 


2 


59 


536 134 643 .4912 


209 639 541 .2743 


874 115 460 .0S07 


1 


60 


.27564 .96126 .28675 3.4874 


•29237 -95630 .30573 3.2709 


.30902 .95106 .32492 3.0777 





Cos. Sin. Cot. Tan. 


Cos. Sin, Cot. Tan. 


Cos. Sin. Cot. Tan. 



740 



73° 



72° 



232 NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
18° 19° 20° 



M. 


Sin. Cos. Tan. Cot. 


Sin. Cos. Tan. Cot, 


Sin. Cos. Tan. Cot, 




o 


.30902 .95106 .32492 3.0777 


•32557 -94552 -34433 2.9042 


.34202 .93969 .36397 2.7475 


60 


i 


929 097 524 .0746 


584 542 465 -9015 


229 959 430 .7450 


59 


2 


957 088 556 .0716 


612 533 498 .8987 


257 949 463 -7425 


58 


3 


985 079 588 .0686 


639 523 530 .8960 


284 939 496 -7400 


57 


4 


.31012 070 621 .0655 


667 514 5 6 3 .8933 


311 929 529 .7376 


56 


5 


.31040 .95061 .32653 3.0625 


.32694 .94504 .34596 2.8905 


•34339 .93919 .365622.7351 


55 


6 


068 052 685 .0595 


722 495 628 .8878 


366 909 595 .7326 


54 


7 


095 °43 7*7 -°5 6 5 


749 485 661 .8851 


393 899 628 .7302 


53 


8 


123 033 749 .0535 


777 476 693 .8824 


421 889 661 .7277 


52 


9 


151 024 782 .0505 


804 466 726 .8797 


448 . 879 694 .7253 


5i 


10 


.31178.95015 .328143.0475 


.32832 .94457 .34758 2.8770 


■34475 .93869 -36727 2.7228 


50 


ii 


206 006 846 .0445 


859 447 79i .8743 


5°3 859 760 .7204 


49 


12 


233 -94997 878 .0415 


8S7 438 824 .8716 


530 849 793 .7179 


48 


13 


261 988 911 .0385 


914 428 856 .8689 


557 839 826 .7155 


47 


14 


289 979 943 .0356 


942 418 889 .8662 


584 829 859 .7130 


46 


15 


.31316 .94970 .32975 3.0326 


.32969 .94409 -34922 2.8636 


.34612 .93819 .36892 2.7106 


45 


16 


344 961 .33007 .0296 


997 399 954 -8609 


639 809 925 .7082 


44 


17 


372 952 040 .0267 


.33024 390 987 .8582 


666 799 958 .7058 


43 


18 


399 943 07 2 -0237 


051 380.35020 .8556 


694 789 991 .7034 


42 


19 


427 933 104 .0208 


079 370 052 .8529 


721 779 .37024 .7009 


4i 


20 


•3H54 -94924 -33136 3.0178 


.33106 .94361 -35085 2.8502 


•34748 .93769 -37°57 2.6985 


40 


21 


482 915 169 .0149 


134 351 118 .8476 


775 759 090 .6961 


39 


22 


510 906 201 .0120 


161 342 150 .8449 


803 748 123 .6937 


38 


23 


537 897 233 .0090 


189 332 183 .8423 


830 738 157 -6913 


37 


24 


565 888 266 .0061 


216 322 216 .8397 


857 728 190 .6889 


36 


25 


•31593 -94878 .332983.0032 


.33244 .94313 .35248 2.8370 


.34884 .93718 .37223 2.6865 


35 


26 


620 869 330 .0003 


271 303 281 .8344 


912 708 256 .6841 


34 


27 


648 860 363 2.9974 


298 293 314 .8318 


939 698 289 .6818 


33 


28 


6 75 851 395 .9945 


326 284 346 .8291 


966 688 322 .6794 


32 


2g 


703 842 427 .9916 


353 274 379 .8265 


993 677 355 .6770 


3i 


30 


.31730 .94832 .33460 2.9887 


.33381 .94264 .35412 2.8239 


.35021 .93667 .37388 2.6746 


30 


3i 


758 823 492 .9858 


408 254 445 .8213 


048 657 422 .6723 


29 


32 


786 814 524 .9829 


436 245 477 .8187 


°75 647 455 .6699 


28 


33 


813 805 557 .9800 


463 235 . 510 .8161 


102 637 488 .6675 


27 


34 


841 795 589 .9772 


490 225 543 .8135 


130 626 521 .6652 


26 


35 


.31868 .94786 .33621 2.9743 


.33518 .94215 .35576 2.8109 


•35 I 57 .93616.375542.6628 


25 


36 


896 777 654 .9714 


545 206 608 .8083 


184 606 588 .6605 


24 


37 


923 768 686 .9686 


573 196 641 .8057 


211 596 621 .6581 


23 


38 


951 758 718 .9657 


600 186 674 .8032 


239 585 654 .6558 


22 


39 


979 749 75 ! -9629 


627 176 707 .8006 


266 575 687 .6534 


21 


40 


.32006 .94740 .33783 2.9600 


•33655 .94167 .35740 2.7980 


•35293.93565 .377202.6511 


20 


4i 


034 730 816 .9572 


682 157 772 .7955 


320 555 754 .6488 


19 


42 


061 721 848 .9544 


710 147 805 .7929 


347 544 787 -6464 


18 


43 


089 712 881 .9515 


737 J 37 838 .7903 


375 534 820 .6441 


17 


44 


116 702 913 .9487 


764 127 871 .7878 


402 524 853 .6418 


16 


45 


.32144 .94693 .33945 2.9459 


.33792 .941 18 .35904 2.7852 


•35429 -935 H -37887 2.6395 


15 


46 


171 684 978 .9431 


819 108 937 .7827 


456 503 920 .6371 


14 


47 


199 674 .34010 .9403 


846 098 969 .7801 


484 493 953 -6348 


13 


48 


227 665 043 .9375 


874 088 .36002 .7776 


511 483 986 .6325 


12 


49 


254 656 075 .9347 


901 078 035 .7751 


538 472 .38020 .6302 


11 


50 


.32282 .94646 .34108 2.9319 


.33929 .94068 .36068 2.7725 


•35565 -93462 .38053 2.6279 


10 


5i 


309 637 140 .9291 


956 058 101 .7700 


592 452 086 .6256 


9 


52 


337 62 7 173 -9263 


983 049 134 .7675 


619 441 120 .6233 


8 


53 


364 618 205 .9235 


.34011 039 167 .7650 


647 431 153 -6210 


7 


54 


392 609 238 .9208 


038 029 199 .7625 


674 420 186 .6187 


6 


55 


.32419 .94599 .34270 2.9180 


.34065 .94019 .36232 2.7600 


.35701 .93410 .38220 2.6165 


5 


56 


447 590 303 .9152 


093 009 265 .7575 


728 400 253 .6142 


4 


57 


474 580 335 -9125 


120.93999 298 .7550 


755 389 286 .6119 


3 


58 


502 571 368 .9097 


147 989 33i 7525 


782 379 320 .6096 


2 


59 


529 561 400 .9070 


175 979 364 .750° 


810 368 353 .6074 


1 


60 


•32557 .94552 -34433 2.9042 


.34202 .93969 .36397 2.7475 


•35837.93358.383862.6051 







Cos. Sin. Cot. Tan. 


Cos. Sin. Cot. Tan. 


Cos. Sin. Cot. Tan. 


M. 



71 c 



70° 



69° 



NATURAL SIXES, COSINES, TANGENTS, AND COTANGENTS. 233 
21° 22P 23° 



K, 


Sin. Cos. Tan. Cot. 


Sin. Cos. Tan. Cot. 


Sin. Cos. Tan. Cot. 







.35 8 37 -9335 s .383862.6051 


.37461 .92718 .40403 2.4751 


.39073 .92050 .42447 2.3559 


60 


i 


864 348 420 .6028 


488 707 436 .4730 


100 039 482 .3539 


59 


2 


891 337 453 -6006 


515 697 470 .4709 


127 028 516 .3520 


58 


3 


918 327 487 .5983 


542 686 504 .4689 


153 016 551 .3501 


57 


4 


945 3i6 520 .5961 


569 675 538 .4668 


180 005 585 .3483 


56 


5 


•35973 -93306 .38553 2.5938 


•37595 -92664 .40572 2.4648 


.39207 .91994 .42619 2.3464 


55 


6 


.36000 295 587 .5916 


622 653 606 .4627 


234 982 654 .3445 


54 


7 


027 285 620 .5893 


649 642 640 .4606 


260 971 688 .3426 


53 


8 


054 274 654 .5871 


676 631 674 .4586 


287 959 722 .3407 


52 


9 


081 264 687 .5848 


703 620 707 .4566 


314 948 757 .3388 


5i 


IO 


.36108 .93253 .38721 2.5826 


.37730 .92609 .40741 2.4545 


.39341 .91936 .42791 2.3369 


50 


ii 


135 2 43 754 .5804 


757 598 775 -4525 


367 925 826 .3351 


49 


12 


162 232 787 .5782 


784 587 809 .4504 


394 914 860 .3332 


48 


13 


190 222 821 .5759 


811 576 843 .4484 


421 902 894 .3313 


47 


14 


217 211 854 .5737 


838 565 877 .4464 


448 891 929 .3294 


46 


15 


.36244 .93201 .38888 2.5715 


.37865 .92554.40911 2.4443 


•39474 .91879 .42963 2.3276 


45 


16 


271 190 921 .5693 


892 543 945 .4423 


501 868 998 .3257 


44 


17 


298 180 955 .5671 


9i9 532 979 4403 


528 856.43032 .3238 


43 


18 


325 169 988 .5649 


946 521 .41013 4383 


555 845 067 .3220 


42 


19 


352 159 -39022 .5627 


973 5 IQ °47 4362 


581 833 101 .3201 


4i 


20 


•36379 -93148 .39055 2.5605 


•37999 -92499 41081 2.4342 


.39608 .91822 .43136 2.3183 


40 


21 


406 137 089 .5583 


.38026 488 115 .4322 


635 810 170 .3164 


39 


22 


434 127 122 .5561 


053 477 149 4302 


661 799 205 .3146 


38 


23 


461 116 156 .5539 


080 466 183 .4282 


688 787 239 .3127 


37 


24 


488 106 190 .5517 


107 455 217 .4262 


7*5 775 274 .3109 


36 


25 


•36515 -93095 -39223 2.5495 


.38134 .92444 4i 25 1 2.4242 


.39741 .91764.433082.3090 


35 


26 


542 084 257 .5473 


161 432 285 .4222 


768 75 2 343 .3072 


34 


27 


569 074 290 .5452 


188 421 319 .4202 


795 74i 378 .3053 


33 


28 


596 063 324 .5430 


215 4io 353 .4182 


822 729 412 .3035 


32 


29 


623 052 357 .5408 


241 399 387 4162 


848 718 447 .3017 


3i 


30 


.36650 .93042 .39391 2.5386 


.38268 .92388 .41421 2.4142 


•39875 -91706 .43481 2.2998 


30 


31 


677 031 425 .5365 


295 377 455 4122 


902 694 516 .2980 


29 


32 


704 020 458 .5343 


322 366 490 .4102 


928 683 550 .2962 


28 


33 


731 010 492 .5322 


349 355 5 2 4 4083 


955 671 585 .2944 


27 


34 


758 .92999 526 .5300 


376 343 558 4063 


982 660 620 .2925 


26 


35 


.36785 .92988 .39559 2.5279 


.38403 .92332 .41592 2.4043 


.40008 .91648 .43654 2.2907 


25 


36 


812 978 593 .5257 


430 321 626 .4023 


035 636 689 .2889 


24 


37 


839 967 626 .5236 


456 310 660 .4004 


062 625 724 .2871 


23 


38 


867 956 660 .5214 


483 299 694 .3984 


088 613 758 .2853 


22 


39 


894 945 694 .5193 


510 287 728 .3964 


115 601 793 .2835 


21 


40 


.36921 .92935 .39727 2.5172 


.38537 .92276 .41763 2.3945 


.40141 .91590 .43828 2.2817 


20 


4i 


948 924 761 .5150 


564 265 797 .3925 


168 578 862 .2799 


19 


42 


975 913 795 -5 I2 9 


591 254 831 .3906 


195 566 897 .2781 


18 


43 


.37002 902 829 .5108 


617 243 865 .38S6 


221 555 932 .2763 


17 


44 


029 892 862 .5086 


644 231 899 .3867 


248 543 966 .2745 


16 


45 


.37056 .92881 .39896 2.5065 


.3S671 .92220 .41933 2.3847 


40275 .91531 44001 2.2727 


15 


46 


083 870 930 .5044 


698 209 968 .3828 


301 519 036 .2709 


14 


47 


no 859 963 .5023 


725 198 .42002 .3808 


328 508 071 .2691 


13 


48 


137 849 997 .5002 


752 186 036 .3789 


355 496 105 .2673 


12 


49 


164 838 .40031 .4981 


778 175 070 .3770 


381 484 140 .2655 


n 


50 


.37191 .92827 .40065 2.4960 


.38805 .92164 .42105 2.3750 


.40408 .9147 2 -44175 2.2637 


10 


5i 


218 816 098 .4939 


832 152 139 -373* 


434 461 210 .2620 


9 


52 


245 805 132 .4918 


859 Hi 173 -3712 


461 449 244 .2602 


8 


53 


272 794 166 .4897 


886 130 207 .3693 


488 437 279 .2584 


7 


54 


299 784 200 .4876 


912 119 242 .3673 


514 425 314 .2566 


6 


55 


.37326 .92773 .40234 2.4855 


.38939 .92107 .42276 2.3654 


.40541 .91414 .44349 2.2549 


5 


56 


353 762 267 .4834 


966 096 310 .3635 


567 402 384 .2531 


4 


57 


380 751 301 .4813 


993 085 345 -3616 


594 39o 418 .2513 


3 


58 


407 740 335 -4792 


.39020 073 379 .3597 


621 378 453 .2496 


2 


59 


434 729 369 -477 2 


046 062 413 .3578 


647 366 488 .2478 


1 


60 


.37461 .92718 .40403 2.4751 


.39073 .92050 .42447 2.3559 


.40674 .91355 .445 2 3 2.2460 







Cos, Sin. Cot. Tan. 


Cos. Sin. Cot. Tan. 


Cos. Sin. Cot. Tan. 


M. 



68° 



67° 



66° 



234 NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
24° 25° 26° 



o 

i 

2 

3 
4 

5 
6 

7 
8 

9 
io 
ii 

12 

13 
14 

15 
16 

17 
18 

19 

20 
21 
22 
23 

24 

25 
26 

27 
28 

29 

30 
31 
32 

33 
34 

35 
36 
37 
38 
39 
40 

4i 
42 

43 
44 

45 
46 

47 
48 

49 
50 
5i 
52 
53 
54 

55 
56 
57 
58 
59 
60 



Sin. Cos. Tan. 



Cot. 



.40674 
700 

727 

780 
.40806 

833 
860 
886 
913 

.40939 
966 
992 

.41019 
045 

.41072 
098 
125 
*5* 
178 

.41204 
231 

257 
284 
310 

41337 
363 
39o 
416 

443 
.41469 
496 
522 
549 
575 
.41602 
628 

655 
681 
707 

•41734 
760 
787 

813 
840 

.41866 
892 
919 

945 
972 

.41998 
.42024 

051 
077 
104 
42130 
156 

183 
209 

235 
42262 



•91355 
343 
33* 
319 
307 

•91295 
283 
272 
260 
248 

.91236 
224 
212 
200 
188 

.91176 
164 
152 

140 
128 

.91116 
104 
092 
080 
068 

.91056 
044 
032 
020 
008 

.90996 

984 
972 
960 
948 
.90936 

924 
911 

899 
887 

90875 
863 
851 

839 
826 



.44523 2.2460 
558 .2443 
593 .2425 
627 .2408 
662 .2390 

.44697 2.2373 
732 .2355 
767 -2338 
802 .2320 

^37 - 2 3°3 
.44872 2.2286 
907 .2268 
942 .2251 
977 -2234 
.45012 .2216 

.45047 2.2199 
082 .2182 
117 .2165 
152 .2148 
187 .2130 

.45222 2.21 13 
257 .2096 
292 .2079 
327 .2062 
362 .2045 

•45397 2.2028 
432 .2011 
467 .1994 
502 .1977 
538 .i960 

•45573 2.1943 
608 .1926 
643 .1909 
678 .1892 
713 .1876 

.45748 2.1859 
784 .1842 
819 .1825 
854 .1808 
889 .1792 

.45924 2.1775 
960 .1758 



90814 
802 



790 
778 
766 

•90753 
741 

729 
717 
704 

.90692 
680 
668 
655 
643 

90631 



995 
.46030 

065 
.46101 

136 

171 

206 

242 
.46277 2, 

312 

348 ■ 

383 • 

418 . 

.46454 2. 
489 • 
525 • 
560 

595 



.1742 

•1725 
.1708 

2.1692 

•^75 

.1659 
.1642 
.1625 
1609 
1592 

*S7^> 
1560 

1543 

*5*7 
1510 

1494 
1478 
1461 



46631 2.1445 



Cos. Sin, 



Cot. 



Tan. 



Sin. Cos. Tan. Cot. 



.42262 
288 
315 
341 

367 

•42394 
420 
446 
473 
499 

•42525 
552 
578 
604 

631 
42657 
683 
709 
736 
762 

42788 

841 
867 

894 



.90631 
618 
606 

594 
582 
.90569 
557 
545 
532 
520 

.90507 
495 
483 
470 

458 
90446 

433 
421 
408 
396 
90383 
37* 
358 
346 
334 



.42920 .90321 
946 
972 

999 
43025 

43051 



.46631 2.1445 
666 .1429 
702 .1413 
737 -1396 
772 .1380 

.46808 2.1364 
843 -1348 
879 .1332 
914 .1315 
950 .1299 

.46985 2.1283 

.47021 .1267 

056 .1251 

092 .1235 

128 .1219 

.47163 2.1203 
199 .1187 
234 .1171 
270 .1155 
305 .1139 

47341 2.1123 
377 -1107 
412 .1092 
448 .1076 
483 .1060 

475 * 9 2.1044 
555 .1028 
59o .1013 
626 .0997 
662 .0081 



Sin. Cos. Tan. Cot. 



309 
296 
284 
271 

90259 .47698 2.0965 
°77 246 733 .0950 
104 233 769 .0934 
130 221 805 .0918 
156 208 840 .0903 
.43182 .90196 .47876 2.0887 
209 183 912 .0872 
235 171 948 .0856 
261 158 984 .0840 
287 146 .48019 .0825 

43313 -90I33 48055 2.0809 
340 120 091 .0794 
366 108 127 .0778 
392 095 163 .0763 
418 082 198 .0748 

43445 .90070 .48234 2.0732 
471 057 270 .0717 

497 °45 3° 6 070 x 
523 032 342 .0686 
549 019 378 .0671 

•43575 -90007 .48414 2.0655 
602 .89994 450 .0640 
628 981 486 .0625 
654 968 521 .0609 
6 8o 95 6 557 -°594 

.43706 .89943 48593 2.0579 
733 93o 629 .0564 
759 918 665 .0549 
785 905 701 .0533 
811 892 737 .0518 

43837 -89879 48773 2.0503 



65° 



Cos. Sin. Cot. Tan. 

64° 



•43837 -89879 .48773 2.0503 
863 867 809 .0488 
889 854 845 .0473 
916 841 881 .0458 
942 828 917 .0443 

.43968 .89816 .48953 2.0428 
994 803 989 .0413 

.44020 790 .49026 .0398 
046 777 062 .0383 
072 764 098 .0368 

.44098 -89752 .49134 2.0353 
124 739 170 .0338 
151 726 206 .0323 

177 7 l 3 242 .0308 
203 700 278 .0293 

.44229 .89687 .49315 2.0278 

255 6 74 35 1 -0263 
281 662 387 .0248 
307 649 423 .0233 
333 636 459 .0219 

.44359 .89623 .49495 2.0204 
385 610 532 .0189 
411 597 5 68 .0174 
437 584 604 .0160 
464 571 640 .0145 

.44490 .89558 .49677 2.0130 

5 l6 545 7*3 -0115 
542 532 749 .0101 
568 519 786 .0086 
594 506 822 .0072 

.44620 .89493 49858 2.0057 
646 480 894 .0042 
672 467 931 .0028 
698 454 967 .0013 
724 441 .50004 1.9999 

.44750 .89428 .50040 1.9984 
776 415. 076 .9970 
802 402 113 .9955 
828 389 149 .9941 
854 376 185 .9926 

.44880 .89363 .50222 1. 9912 
906 350 258 .9897 
932 337 295 .9883 
958 324 33* -9868 
984 311 368 .9854 

.45010 .89298 .50404 1.9840 
036 285 441 .9825 
062 272 477 .9811 
088 259 514 .9797 
114 245 550 .9782 

.45140 .89232 .50587 1.9768 
166 219 623 .9754 
1.92 206 660 .9740 
218 193 696 .9725 
243 180 733 .9711 

45269 .89167 .50769 1.9697 
295 153 806 .9683 
321 140 843 .9669 
347 127 879 .9654 
373 "4 916 .9640 

45399 -89101 .50953 1.9626 



60 

59 
58 
57 
56 

55 
54 
53 
52 
5i 
50 
49 
48 
47 
46 

45 
44 
43 
42 

41 
40 
39 
38 
37 
36 

35 
34 
33 
32 
3i 
30 
29 
28 
27 
26 

25 
24 
23 
22 
21 



Cos. Sin. Cot. Tan. M 

63° 



NATURAL SIXES, COSINES, TANGENTS, AND COTANGENTS. 235 
27° 28° 29° 



M. 


Sin. Cos. Tan. Cot. 


Sin. Cos. Tan. Cot. 


Sin. Cos. Tan, Cot. 




o 


•45399 -89I 01 -5°953 1-9626 


.46947 .88295 .53171 1.8807 


.48481 .87462 .55431 1.8040 


60 


i 


425 087 989 .9612 


973 281 208 .8794 


506 448 469 .8028 


59 


2 


451 074.51026 .9598 


999 267 246 .8781 


532 434 5°7 -8016 


58 


3 


477 061 063 ,9584 


.47024 254 283 .8768 


557 420 545 -8003 


57 


4 


503 048 099 .9570 


050 240 320 .8755 


583 406 583 .7991 


56 


5 


.45529 .89035 .51136 1.9556 


.47076 .88226 .53358 1.8741 


.48608 .87391 .55621 1.7979 


55 


6 


554 021 173 .9542 


101 213 395 .8728 


634 377 6 59 -7966 


54 


7 


580 008 209 .9528 


127 199 432 .8715 


659 363 697 .7954 


53 


8 


606 .88995 246 .9514 


153 185 470 .8702 


684 349 736 .7942 


52 


9 


632 981 283 .9500 


178 172 507 .8689 


710 335 774 -793o 


5i 


IO 


.45658 .88968 .51319 1.9486 


.47204 .88158 .53545 1.8676 


.48735 .87321 .55812 1. 7917 


50 


ii 


684 955 356 .9472 


229 144 582 .8663 


761 306 850 .7905 


49 


12 


710 942 393 .9458 


255 130 620 .8650 


786 292 888 .7893 


48 


13 


736 928 430 .9444 


281 117 657 .8637 


811 278 926 .7881 


47 


14 


762 915 467 .9430 


306 103 694 .8624 


837 264 964 .7868 


46 


15 


.45787 .88902 .51503 1.9416 


.47332 .88089 .53732 1.8611 


.48862 .87250 .56003 1.7856 


45 


16 


813 888 540 .9402 


358 075 769 .8598 


888 235 041 .7844 


44 


17 


839 875 577 .9388 


383 062 807 .8585 


913 221 079 .7832 


43 


18 


865 862 614 .9375 


409 048 844 .8572 


938 207 117 .7820 


42 


19 


891 848 651 .9361 


434 034 882 .8559 


964 193 156 .7808 


4i 


20 


.45917 .88835 .51688 1.9347 


.47460 .88020 .53920 1.8546 


.48989 .87178 .56194 1.7796 


40 


21 


942 822 724 .9333 


486 006 957 .8533 


.49014 164 232 .7783 


39 


22" 


968 808 761 .9319 


511 .87993 995 .8520 


040 150 270 .7771 


38 


23 


994 795 798 .9306 


537 979 -54032 .8507 


065 136 309 .7759 


37 


24 


.46020 782 835 .9292 


562 965 070 .8495 


090 121 347 .7747 


36 


25 


.46046 .88768 .51872 1.9278 


.47588 .87951 .54107 1.8482 


.49116.87107 .56385 1.7735 


35 


26 


°7 2 755 909 -9265 


614 937 145 -8469 


141 093 424 .7723 


34 


27 


097 741 946 .9251 


639 923 183 .8456 


166 079 462 .7711 


33 


28 


123 728 983 .9237 


665 9Q9 220 .8443 


192 064 501 .7699 


32 


29 


149 715 .52020 .9223 


690 896 258 .8430 


217 050 539 -7687 


3i 


30 


.46175 .88701 .52057 1.9210 


.47716 .87882 .54296 1. 8418 


.49242 .87036 .56577 1.7675 


30 


31 


201 688 094 .9196 


741 868 333 .8405 


268 021 616 .7663 


29 


32 


226 674 131 .9183 


7 6 7 854 371 .8392 


293 o°7 6 54 •7 6 5 I 


28 


33 


252 661 168 .9169 


793 840 409 .8379 


318.86993 693 .7639 


27 


34 


278 647 205 .9155 


818 826 446 .8367 


344 978 73 1 -7627 


26 


35 


.46304 .88634 .52242 1. 9142 


.47844 .87812 .54484 1.8354 


.49369 .86964 .56769 1. 7615 


25 


36 


330 620' 279 .9128 


869 798 522 .8341 


394 949 808 .7603 


24 


37 


355 6o 7 3i6 .9115 


895 784 5 6 ° -8329 


419 935 846 .7591 


23 


38 


381 593 353 -9ioi 


920 770 597 .8316 


445 92i 885 .7579 


22 


39 


407 580 390 .9088 


946 756 635 .8303 


470 906 923 .7567 


21 


40 


.46433 .88566 .52427 1.9074 


.47971 .87743 .54673 1.8291 


.49495 .86892 .56962 1.7556 


20 


4i 


458 553 464 -9061 


997 729 711 .8278 


521 878 .57000 .7544 


19 


42 


484 539 5 QI -9047 


.48022 715 748 .8265 


546 863 039 .7532 


18 


43 


510 526 538 .9034 


048 701 786 .8253 


571 849 078 .7520 


17 


44 


536 5 12 575 -9020 


073 687 824 .8240 


596 834 116 .7508 


16 


45 


.46561 .88499 .52613 1.9007 


.48099 .87673 .54862 1.8228 


.49622 .86820 .57155 1-7496 


15 


46 


587 485 650 .8993 


124 659 900 .8215 


647 805 193 .7485 


14 


47 


613 472 687 .8980 


150 645 938' .8202 


672 791 232 .7473 


13 


48 


639 458 724 -8967 


175 6 3i 975 -8190 


697 777 271 .7461 


12 


49 


664 445 761 .8953 


201 617 .55013 .8177 


723 7 62 309 -7449 


11 


50 


.46690 .88431 .52798 1.8940 


.48226 .87603 .55051 1.8165 


.49748 .86748 .57348 1.7437 


10 


51 


716 417 836 .8927 


252 589 089 .8152 


773 733 386 .7426 


9 


52 


742 404 873 .8913 


277 575 I2 7 -8140 


798 719 425 -74H 


8 


53 


767 390 910 .8900 


303 561 165 .8127 


824 704 464 .7402 


7 


54 


793 377 947 -8887 


328 546 203 .8115 


849 690 503 .7391 


6 


55 


.46819 .88363 .52985 1.8873 


•48354 -87532 .55241 1-8103 


.49874.86675 .57541 1.7379 


5 


56 


844 349 .53022 .8860 


379 518 279 .8090 


899 661 580 .7367 


4 


57 


870 336 059 .8847 


405 504 317 .8078 


924 646 619 .7355 


3 


58 


896 322 096 .8834 


430 490 355 -8065 


950 632 657 .7344 


2 


59 


921 308 134 .8820 


456 476 393 .8053 


975 617 696 .7332 


1 


60 


.46947 -88295 .53171 1.8807 


.48481 .87462 .55431 1.8040 


.50000 .86603 .57735 i-73 21 







Cos. Sin. Cot. Tan. 


Cos. Sin. Cot. Tan. 


Cos. Sin. Cot. Tan. 


M. 



62° 



61° 



60° 



236 NATURAL SINES, COSINES, TANGENTS, 
30° 31° 



AND COTANGENTS. 
32° 



M. 


Sin. Cos. Tan. 


Cot. 


Sin. Cos. Tan. Cot. 


Sin, Cos. Tan. Cot. 




o 


.50000 .86603 .57735 


1.7321 


.51504 .85717 .60086 1.6643 


.52992 .84805 .62487 1.6003 


60 


i 


° 2 5 5 88 774 


•7309 


529 702 126 .6632 


.53017 789 5 2 7 -5993 


59 


2 


050 573 813 


.7297 


554 687 165 .6621 


041 774 568 .5983 


58 


3 


°7 6 559 851 


.7286 


579 672 205 .6610 


066 759 608 .5972 


57 


4 


101 544 890 


.7274 


604 657 245 .6599 


091 743 649 .5962 


56 


5 


.50126 .86530 .57929 


1.7262 


.51628 .85642 .60284 1.6588 


.53115 .84728.62689 1.5952 


55 


6 


151 515 968 


•7 2 5 J 


653 627 324 .6577 


140 712 730 .5941 


54 


7 


176 501 .58007 


•7 2 39 


678 612 364 .6566 


164 697 770 .5931 


53 


8 


201 486 046 


.7228 


7°3 597 403 -6555 


189 681 811 .5921 


52 


9 


227 471 085 


.7216 


728 582 443 .6545 


214 666 852 .5911 


5i 


10 


.50252 .86457 -5 8l2 4 


1.7205 


•5 I 753-855 6 7- 6 °483 1-6534 


.53238 .84650 .62892 1.5900 


50 


ii 


277 442 162 


•7*93 


778 55 1 5 22 - 6 5 2 3 


2 63 635 933 .5890 


49 


12 


302 427 201 


.7182 


803 536 5 62 - 6 5 12 


288 619 973 .5880 


48 


13 


327 413 240 


.7170 


828 521 602 .6501 


312 604.63014 .5869 


47 


H 


352 398 279 


•7*59 


852 506 642 .6490 


337 538 055 .5859 


46 


15 


.50377 .86384 .58318 


1. 7147 


.51877 .85491 .60681 1.6479 


.53361 .84573 .63095 1.5849 


45 


16 


403 369 357 


•7*36 


902 476 721 .6469 


386 557 136 .5839 


44 


17 


428 354 396 


.7124 


927 461 761 .6458 


411 542 177 .5829 


43 


18 


453 340 435 


■7 ll 3 


952 446 801 .6447 


435 5 26 2I 7 -5818 


42 


19 


478 325 474 


.7102 


c> 77 431 841 .6436 


460 511 258 .5808 


4i 


20 


.50503 .86310.58513 


[.7090 


.52002 .85416 .60881 1.6426 


.53484 .84495 - 6 3 2 99 1-5798 


40 


21 


5 28 2 95 55 2 


.7079 


026 401 921 .6415 


509 480 340 .5788 


39 


22 


553 28 i 59i 


.7067 


°5 I 385 960 .6404 


534 464 380 .5778 


38 


23 


578 266 631 


.7056 


076 370 .61000 .6393 


558 448 421 .5768 


37 


24 


603 251 670 


.7045 


101 355 040 .6383 


583 433 462 .5757 


36 


25 


.50628 .86237 .58709 


t-7°33 


.52126 .85340 .61080 1.6372 


.53607 .84417 -63503 1-5747 


35 


26 


654 222 748 


.7022 


151 325 120 .6361 


632 402 544 .5737 


34 


27 


679 207 787 


.7011 


175 310 160 .6351 


656 386 584 .5727 


33 


28 


704 192 826 


.6999 


200 294 200 .6340 


681 370 625 .5717 


32 


29 


729 178 865 


.6988 


225 279 240 .6329 


705 355 666 .5707 


3i 


30 


.50754 .86163 .58905 


[.6977 


.52250 .85264 .61280 1. 6319 


•5373o .84339 -63707 I-5697 


30 


31 


779 H8 944 


.6965 


2 75 2 49 3 2 ° - 6 3o8 


754 3 2 4 748 .5687 


29 


32 


804 133 983 


•6954 


2 99 2 34 3 6 ° - 62 97 


779 308 789 .5677 


28 


33 


829 119 .59022 


.6943 


324 218 -400 .6287 


804 292 830 .5667 


27 


34 


854 104 061 


.6932 


349 2 °3 44o .6276 


828 277 871 .5657 


26 


35 


.50879 .86089 .59101 


[.6920 


.52374 .85188 .61480 1.6265 


.53853 .84261 .63912 1.5647 


25 


36 


904 074 140 


.6909 


399 173 5 2 ° - 62 55 


^77 2 45 953 -5637 


24 


37 


929 059 179 


.6898 


423 157 561 .6244 


902 230 994 .5627 


23 


38 


954 045 218 


.6887 


448 142 601 .6234 


926 214 .64035 .5617 


22 


39 


979 030 258 


.6875 


473 127 641 .6223 


951 198 076 .5607 


21 


40 


.51004 .86015 .59297 


[.6864 


.52498 .85112 .61681 1.6212 


•53975 .84182.64117 1.5597 


20 


4i 


029 000 336 


.6853 


522 096 721 .6202 


.54000 167 i$8 .5587 


19 


42 


054 -85985 376 


.6842 


547 081 761 .6191 


024 151 199 .5577 


18 


43 


079 970 415 


.6831 


572 066 801 .6181 


049 135 2 4Q .5567 


17 


44 


104 956 454 


.6820 


597 °5 X 842 .6170 


073 120 281 .5557 


16 


45 


.51129 .85941 .59494 


[.6808 


.52621 .85035 .61882 1. 6160 


.54097 .84104 .64322 1.5547 


15 


46 


154 926 533 


.6797 


646 020 922 .6149 


122 088 363 .5537 


14 


47 


179 9ii 573 


.6786 


671 005 962 .6139 


146 072 404 .5527 


13 


48 


204 896 612 


•6775 


696 .84989 .62003 .6128 


171 057 446 .5517 


12 


49 


229 881 651 


.6764 


720 974 043 .6118 


195 041 487 .5507 


11 


50 


.51254 .85866 .59691 ] 


•6753 


.52745 .84959 .62083 1.6107 


.54220 .84025 .64528 1.5497 


10 


5i 


2 79 851 / 730 


.6742 


770 943 124 .6097 


244 009 569 .5487 


9 


52 


304 836 770 


•6731 


794 928 164 .6087 


269 .83994 610 .5477 


8 


53 


329 821 809 


.6720 


.819 913 204 .6076 


2 93 978 652 .5468 


7 


54 


354 806 849 


.6709 


844 897 245 .6066 


317 962 693 .5458 


6 


55 


•5 r 379 -85792 .59888 ] 


.6698 


.52869 .84882 .62285 1.6055 


.54342 .83946 .64734 1.5448 


5 


56 


404 777 928 


.6687 


893 866 325 .6045 


366 930 775 .5438 


4 


57 


429 762 967 


.6676 


918 851 366 .6034 


391 915 817 .5428 


3 


58 


454 747 .60007 


.6665 


943 836 406 .6024 


415 899 858 .5418 


2 


59 


479 732 046 


.6654 


967 820 446 .6014 


440 883 899 .5408 


1 


60 


.51504 .85717 .60086 ] 


.6643 


.52992 .84805 .62487 1.6003 


.54464 .83867 .64941 1.5399 









Cos. Sin. Cot. 


Tan. 


Cos. Sin. Cot, Tan. 


Cos. Sin. Cot. Tan. 


M. J 



59° 



58° 



57° 



NATURAL SINES, COSINES, TANGENTS, 
33° 34° 



AND COTANGENTS. 237 
35° 



M. 
o 


Sin. Cos. Tan. Cot. 


Sin. Cos, Tan. Cot. 


Sin. Cos. Tan. 


Cot. 




.54464 .83867 .64941 1.5399 


•559 J 9 -82904 .67451 1.4826 


•57358 .81915 -70021 


1.42S1 


eo 


i 


488 851 982 .5389 


943 887 493 .4816 


381 899 064 


4273 


59 


2 


513 835 .65024 .5379 


968 871 536 .4807 


405 882 107 


.4264 


58 


3 


537 819 065 .5309 


992 855 578 .4798 


429 865 151 


4255 


57 


4 


561 804 106 .5359 


.56016 839 620 .4788 


453 848 194 


.4246 


56 


5 


.54586.83788.651481.5350 


.56040 .82822 .67663 14779 


.57477 .81832 .70238 


14237 


55 


6 


610 772 189 .5340 


064 806 705 4770 


501 815 281 


.4229 


54 


7 


635 75 6 231 -533° 


088 790 748 .4761 


524 798 325 


.4220 


53 


8 


659 740 272 .5320 


112 773 790 .4751 


548 782 368 


.4211 


52 


9 


683 724 314 .5311 


136 757 832 .4742 


572 765 412 


.4202 


5i 


10 


.54708 .83708 .65355 1.5301 


.56160 .82741 .67875 1.4733 


.57596.81748.70455 


f4i93 


50 


ii 


732 692 397 .5291 


184 724 917 .4724 


6i9 73 1 499 


.4185 


49 


12 


756 676 438 .5282 


208 708 960 .4715 


643 714 542 


.4176 


48 


13 


781 660 480 .5272 


232 692 .6S002 .4705 


667 698 586 


.4167 


47 


14 


805 645 521 .5262 


256 675 045 .4696 


691 681 629 


.4158 


46 


15 


.54829 .83629 .65563 1.5253 


.56280 .82659 .68088 1.4687 


.57715 .81664 .70673 


[.4150 


45 


16 


854 613 604 .5243 


305 643 130 .4678 


738 647 717 


.4141 


44 


17 


878 597 646 .5233 


329 626 173 .4669 


762 631 760 


.4132 


43 


18 


902 581 688 -.5224 


353 610 215 .4659 


786 614 804 


.4124 


42 


19 


927 565 729 .5214 


377 593 258 .4650 


810 597 848 


4115 


4i 


20 


.54951 .83549 .65771 1.5204 


.56401 .82577 .68301 1.4641 


.57833 .81580 .70891 


[.4106 


40 


21 


975 533 813 .5195 


425 5 61 343 4632 


857 5 6 3 935 


.4097 


39 


22 


999 5 J 7 8 54 -5 l8 5 


449 544 386 4623 


881 546 979 


.4089 


38 


23 


.55024 501 896 .5175 


473 528 429 4614 


904 53o .71023 


.4080 


37 


24 


048 485 938 .5166 


497 5 11 47 1 4605 


928 513 066 


.4071 


36 


25 


.55072 .83469 .65980 1.5156 


.56521 .82495 .68514 1.4596 


.57952 .81496 .71110 


[.4063 


35 


26 


•097 453.66021 .5147 


545 478 557 4586 


976 479 154 


4054 


34 


27 


121 437 063 .5137 


569 462 600 .4577 


999 462 198 


4045 


33 


28 


145 421 105 .5127 


593 446 642 .4568 


.58023 445 242 


4037 


32 


29 


169 405 147 .5118 


617 429 685 .4559 


047 428 285 


.4028 


3i 


30 


.55194 .83389 .66189 1. 5108 


.56641 .82413 .68728 14550 


.58070 .81412 .71329 


[.4019 


30 


31 


218 373 230 .5099 


665 396 771 454i 


094 395 373 


.4011 


29 


32 


242 356 272 .5089 


689 380 814 .4532 


118 378 417 


.4002 


28 


33 


266 340 314 .5080 


7 l 3 363 857 .4523 


141 361 461 


•3994 


27 


34 


291 324 35 6 -5°7 


736 347 900 .4514 


165 344 505 


.3985 


26 


35 


•553*5 - 8 33o8 .66398 1.5061 


.56760 .82330 .68942 1.4505 


.58189 .S1327 .71549 


[.3976 


25 


36 


339 292 440 .5051 


784 314 985 .4496 


212 310 593 


•3968 


24 


37 


363 276 482 .5042 


808 297 .69028 .4487 


236 293 637 


•3959 


23 


3S 


388 260 524 .5032 


832 281 071 .4478 


260 276 681 


•395 1 


22 


39 


412 244 566 .5023 


856 264 114 4469 


283 259 725 


•3942 


21 


40 


.55436 .83228 .66608 1. 5013 


.56880 .82248 .69157 1.4460 


.58307 .81242 .71769 


'•3934 


20 


4i 


460 212 650 .5004 


904 231 200 .4451 


330 225 813 


•3925 


19 


42 


484 195 692 .4994 


928 214 243 .4442 


354 208 857 


.3916 


18 


43 


509 179 734 .4985 


952 198 286 .4433 


378 191 901 


.3908 


17 


44 


533 163 776 .4975 


976 181 329 .4424 


401 174 946 


•3899 


16 


45 


.55557 .83147 .66818 1.4966 


.57000.82165 .69372 1.4415 


.58425 .81157 .71990 


[.3891 


15 


46 


581 131 860 .4957 


024 148 416 .4406 


449 HO .72034 


.3882 


14 


47 


605 115 902 .4947 


047 l 3 2 459 4397 


472 123 078 


•3874 


13 


48 


630 098 944 4938 


071 115 502 .4388 


496 106 122 


.3865 


12 


49 


654 082 986 .4928 


095 098 545 .4379 


519 089 167 


.3857 


11 


50 


.55678 .83066 .67028 1.4919 


.57119 .82082 .69588 1.4370 


.58543 .81072.72211 


[.3848 


10 


5i 


702 050 071 .4910 


143 065 631 .4361 


567 055 255 


.3840 


9 


52 


726 034 • 113 .4900 


167 048 675 .4352 


590 038 299 


.3831 


8 


53 


75° OI 7 155 4891 


191 032 718 .4344 


614 021 344 


•3823 


7 


54 


775 001 197 .4882 


215 015 761 .4335 


637 004 388 


.3814 


6 


55 


55799 .82985 .67239 1.4872 


.57238 .81999 .69804 1.4326 


.58661 .80987 .72432 


1.3806 


5 


56 


823 969 282 .4863 


262 982 847 .4317 


684 970 477 


•3798 


4 


57 


847 953 3 2 4 4854 


286 965 891 .4308 


708 953 521 


.3789 


3 


58 


871 936 366 .4844 


310 949 934 4299 


73 1 93° 5°5 


•378i 


2 


59 


895 920 409 .4835 


334 932 977 4290 


755 9i9 610 


•3772 


1 


60 


.55919 .82904 .67451 1.4826 


.57358 .81915 .70021 1.4281 


.58779 .80902 .72654 


i.37 6 4 







Cos. Sin. Cot. Tan. 


Cos. Sin. Cot. Tan. 


Cos. Sin. Cot. 


Tan. 


M. 



56^ 



55 



54; 



238 NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
36° 37° 38° 



M. 




Sin. Cos. Tan. 


Cot. 


Sin. Cos. Tan. Cot. 


Sin. Cos. Tan, Cot. 


60 


.58779 .80902 .72654 


I-3764 


.60182 .79864 .75355 1.3270 


.61566 .78801 .78129 1.2799 


i 


802 885 699 


•3755 


205 846 401 .3262 


589 783 175 .2792 


59 


2 


826 867 743 


•3747 


228 829 447 .3254 


612 765 222 .2784 


58 


3 


849 850 788 


•3739 


251 811 492 .3246 


635 747 269 .2776 


57 


4 


873 &33 832 


.3730 


274 793 538 .3238 


658 729 316 .2769 


56 


5 


.58896 .80816 .72877 


1.3722 


.60298 .79776 .75584 1.3230 


.61681 .78711 .78363 1. 2761 


55 


6 


920 799 921 


•37*3 


321 758 629 .3222 


704 694 410 .2753 


54 


7 


943 782 966 


•3705 


344 741 675 .3214 


726 676 457 .2746 


53 


8 


967 765 .73010 


.3697 


367 723 721 .3206 


749 658 504 .2738 


52 


9 


990 748 055 


.3688 


390 706 767 .3198 


772 640 551 .2731 


5i 


IO 


.59014 .80730 .73100 


[.3680 


.60414 .79688 .75812 1. 3190 


.61795 .78622 .78598 1.2723 


50 


ii 


°37 7*3 144 


.3672 


437 671 858 .3182 


818 604 645 .2715 


49 


12 


061 696 189 


•3663 


460 653 904 .3175 


841 586 692 .2708 


48 


13 


084 679 234 


•3655 


483 635 950 .3167 


864 568 739 .2700 


47 


14 


108 662 278 


•3647 


506 618 996 .3159 


^ 550 786 .2693 


46 


15 


.59131 .80644 .73323 


[.3638 


.60529 .79600 .76042 1.3151 


.61909 .78532 .78834 1.2685 


45 


16 


154 627 368 


•3630 


553 583 088 .3143 


932 514 881 .2677 


44 


17 


178 610 413 


.3622 


576 565 134 .3135 


955 49§ 928 .2670 


43 


18 


201 593 457 


.3613 


599 547 180 .3127 


978 47& 975 .2662 


42 


ig 


22 5 57 6 5° 2 


•3605 


622 530 226 .3119 


.62001 460 .79022 .2655 


4i 


20 


.59248 .80558 .73547 


[-3597 


.60645 .79512 .76272 1.3111 


.62024 -78442 .79070 1.2647 


40 


21 


272 541 592 


.3588 


668 494 318 .3103 


046 424 117 .2640 


39 


22 


295 5 2 4 637 


.3580 


691 477 364 .3095 


069 405 164 .2632 


38 


23 


318 507 681 


•3572 


714 459 410 .3087 


092 387 212 .2624 


37 


24 


342 489 726 


•3564 


738 441 456 -3°79 


115 369 259 .2617 


36 


25 


.59365 .80472 .73771 


1-3555 


.60761 .79424 .76502 1.3072 


.62138 .78351 .79306 1.2609 


35 


26 


389 455 816 


•3547 


784 406 548 .3064 


160 2>33 354 -2602 


34 


27 


412 438 861 


•3539 


807 388 594 .3056 


183 315 401 .2594 


33 


28 


436 420 906 


•3531 


830 371 640 .3048 


206 297 449 .2587 


32 


29 


459 403 95 l 


.3522 


853 353 686 .3040 


229 279 496 .2579 


3i 


30 


.59482 .80386 .73996 


f-35H 


.60876 .79335 .76733 1.3032 


.62251 .78261 .79544 1.2572 


30 


31 


506 368 .74041 


.3506 


899 3*8 779 -3024 


274 243 591 .2564 


29 


32 


529 351 086 


.3498 


922 300 825 .3017 


297 225 639 .2557 


28 


33 


552 334 131 


•3490 


945 282 - 871 .3009 


320 206 686 .2549 


27 


34 


576 316 176 


.3481 


968 264 918 .3001 


342 188 734 .2542 


26 


35 


.59599 .80299 .74221 


t-3473 


.60991 .79247 .76964 1.2993 


.62365 .78170.79781 1.2534 


25 


36 


622 282 267 


•3465 


.61015 229 .77010 .2985 


388 152 829 .2527 


24 


37 


646 264 312 


•3457 


038 211 057 .2977 


411 134 877 .2519 


23 


38 


669 247 357 


•3449 


061 193 103 .2970 


433 116 924 .2512 


22 


39 


693 230 402 


•3440 


084 176 149 .2962 


456 098 972 .2504 


21 


40 


.59716 .80212 .74447 


[-3432 


.61107 .79158 .77196 1.2954 


.62479 .78079 .80020 1.2497 


20 


4i 


739 195 492 


•3424 


130 140 242 .2946 


502 061 067 .2489 


19 


42 


763 178 538 


.3416 


153 122 289 .2938 


524 043 115 .2482 


18 


43 


786 160 583 


.3408 


176 105 335 .2931 


547 ° 2 5 l6 3 -2475 


17 


44 


809 143 628 


.3400 


199 087 382 .2923 


570 007 211 .2467 


16 


45 


.59832 .80125 .74674 


[-3392 


.61222 .79069 .77428 1. 291 5 


.62592 .77988 .80258 1.2460 


15 


46 


856 108 719 


•3384 


245 °5! 475 -2907 


615 970 306 .2452 


14 


47 


879 091 764 


•3375 


268 033 521 .2900 


638 952 354 -2445 


13 


48 


902 073 810 


•3367 


291 016 568 .2892 


660 934 402 .2437 


12 


49 


926 056 855 


•3359 


314 .78998 615 .2884 


683 916 450 .2430 


11 


50 


.59949 .80038 .74900 


[-335 1 


•61337 .78980 .77661 1.2876 


.62706 .77897 .80498 1.2423 


10 


5i 


972 021 946 


•3343 


360 962 708 .2869 


728 879 546 .2415 


9 


52 


995 °°3 99i 


•3335 


383 944 754 -2861 


751 861 594 .2408 


8 


53 


.60019 .79986 .75037 


•3327 


406 926 801 .2853 


774 843 642 .2401 


7 


54 


042 968 082 


•3319 


429 908 848 .2846 


796 824 690 .2393 


6 


55 


.60065 .79951 .75128 


[-3311 


.61451 .78891 .77895 1.2838 


.62819 .77806 .80738 1.2386 


5 


56 


089 934 173 


•3303 


474 873 941 .2830 


842 788 786 .2378 


4 


57 


112 916 219 


.3295 


497 855 988 .2822 


864 769 834 .2371 


3 


58 


135 899 264 


•3287 


520 837 .78035 .2815 


887 751 882 .2364 


2 


59 


158 881 310 


.3278 


543 819 082 .2807 


909 733 93o -2356 


1 


60 


.60182 .79864 .75355 i 


.3270 


.61566 .78801 .78129 1.2799 


.62932 .77715 .80978 1.2349 







Cos. Sin. Cot. 


Tan. 


Cos. Sin. Cot. Tan. 


Cos. Sin. Cot. Tan. 


M. 



53° 



52 ( 



51° 



NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 239 
39° 40° 41° 



M. 


Sin, Cos. Tan. Cot. 


Sin. Cos. Tan. Cot. 


Sin. Cos. Tan. 


Cot, 




o 


.62932 .77715 .80978 1.2349 


.64279 .76604 .83910 1.] 


918 


.65606 .75471 .86929 1 


1504 


60 


i 


955 696 .810.27 .2342 


301 586 960 j 


910 


628 452 980 


1497 


59 


2 


977 678 075 .2334 


323 567 .84009 j 


903 


6 5° 433 -87031 


1490 


58 


3 


.63000 660 123 .2327 


346 548 059 J 


896 


672 414 082 


H83 


57 


4 


022 641 171 .2320 


368 530 108 .] 


889 


694 395 ! 33 


H77 


56 


5 


•63045 -77 62 3 -81220 1.2312 


.64390.76511 .84158 i.i 


882 


•65716.75375 -87184 1 


1470 


55 


6 


068 605 268 .2305 


412 492 208 . 


875 


738 356 236 


1463 


54 


7 


090 586 316 .2298 


435 473 258 . 


868 


759 337 287 


H56 


53 


8 


113. 568 364 .2290 


457 455 307 • 


861 


781 318 338 


H50 


52 


9 


135 55° 4i3 -2283 


479 436 357 - ] 


854 


803 299 389 


1443 


5i 


10 


•63158 -7753 1 -81461 1.2276 


.64501 .76417 .84407 1.] 


847 


.65825 .75280 .87441 1 


1436 


50 


ii 


180 513 510 .2268 


524 398 457 .] 


840 


847 261 492 


H3o 


49 


12 


203 494 55 8 - 22 6i 


546 380 507 .] 


833 


869 241 543 


1423 


48 


13 


225 476 606 .2254 


568 361 556 j 


826 


891 222 5.95 


1416 


47 


14 


248 458 655 .2247 


590 342 606 .1 


819 


913 203 646 


1410 


46 


15 


.63271 .77439 .81703 1.2239 


.64612 .76323 .84656 i.i 


812 


•65935 -75 l8 4 .87698 1 


1403 


45 


16 


293 421 752 .2232 


635 3°4 7° 6 •> 


806 


956 165 749 


1396 


44 


17 


316 402 800 .2225 


657 286 756 .] 


799 


978 146 801 


1389 


43 


1 8 


338 384 849 .2218 


679 267 806 .] 


792 


.66000 126 852 


1383 


42 


19 


361 366 898 .2210 


701 248 856 .] 


785 


022 107 904 


1376 


4i 


20 


•63383 -77347 -81946 1.2203 


.64723 .76229 .84906 i.i 


778 


.66044 .75088 .87955 1 


1369 


40 


21 


406 329 995 .2196 


746 210 956 j 


771 


066 069 .88007 


1363 


39 


22 


428 310 .82044 .2189 


768 192 .85006 .] 


764 


088 050 059 


1356 


38 


23 


451 292 092 .2181 


790 173 o57 • 


757 


109 030 no 


: 349 


37 


24 


473 273 141 .2174 


812 154 107 . 


75o 


131 on 162 


1343 


36 


25 


.63496 .77255 .82190 1. 2167 


.64834.76135 •85 I 57 i- 


743 


.66153 .74992 .88214 I 


1336 


35 


26 


518 236 238 .2160 


856 116 207 . 


736 


175 973 265 


1329 


34 


27 


540 218 287 .2153 


878 097 257 . 


729 


197 953 3i7 


1323 


33 


28 


563 199 336 .2145 


901 078 308 . 


722 


218 934 369 


1316 


32 


29 


585 181 385 .2138 


923 059 358 . 


715 


240 915 421 


1310 


31 


30 


.63608 .77162 .82434 1.2131 


.64945 -76041 .85408 1. 


708 


.66262 .74896 .88473 l 


1303 


30 


31 


630 144 483 .2124 


967 022 458 . 


702 


284 876 524 


1296 


29 


32 


653 125 531 .2117 


989 003 509 . 


695 


306 857 576 


1290 


28 


33 


675 io 7 580 .2109 


.65011 .75984 559 . 


[688 


327 838 628 


1283 


27 


34 


698 088 629 .2102 


033 965 609 . 


[681 


349 818 680 


1276 


26 


35 


.63720 .77070 .82678 1.2095 


.65055 .75946 .85660 1. 


[674 


•66371 -74799 -88732 1 


1270 


25 


36 


742 051 727 .2088 


077 927 710 . 


[667 


393 780 784 


.1263 


24 


37 


765 033 776 .2081 


100 908 761 . 


[660 


414 760 836 


1257 


23 


38 


787 014 825 .2074 


122 889 811 . 


1653 


436 741 8S8 


1250 


22 


39 


810 .76996 874 .2066 


144 870 862 . 


[647 


458 722 940 


1243 


21 


40 


.63832 .76977 .82923 1.2059 


.65166 .75851 .85912 1. 


[640 


.66480 .74703 .88992 1 


1237 


20 


4i 


854 959 972 .2052 


188 832 963 . 


'6 3 3 


501 683 .89045 


.1230 


19 


42 


877 940 .83022 .2045 


210 813 .86014 . 


1626 


523 664 097 


.1224 


18 


43 


899 921 071 .2038 


232 794 064 . 


[619 


545 6 44 149 


1217 


17 


44 


922 903 120 .2031 


254 775 "5 • 


[612 


566 625 201 


1211 


16 


45 


.63944 .76884 .83169 1.2024 


.65276 .75756 .86166 1. 


1606 


.66588 .74606 .89253 1 


1204 


15 


46 


966 866 218 .2017 


298 738 216 . 


'599 


610 586 306 


1197 


14 


47 


989 847 268 .2009 


320 719 267 . 


1592 


632 567 358 


.1191 


13 


48 


.64011 828 317 .2002 


342 700 318 . 


.585 


653 548 410 


.1184 


12 


49 


033 810 366 .1995 


364 680 368 . 


'578 


675 528 463 


.1178 


11 


50 


.64056 .76791 .83415 1.1988 


.65386 75661 .86419 1. 


'571 


.66697 .74509 .89515 1 


.1171 


10 


5i 


078 772 465 .1981 


408 642 470 . 


[ 5 6 5 


718 489 567 


.1165 


9 


52 


100 754 514 .1974 


430 623 521 . 


[ 55 8 


740 470 620 


1158 


8 


53 


123 735 564 -1967 


452 604 572 . 


[ 55* 


762 451 672 


1152 


7 


54 


145 717 613 .i960 


474 585 623 . 


'544 


783 43i 725 


1145 


6 


55 


.64167 .76698 .83662 1. 1953 


.65496 .75566 .86674 1. 


'538 


.66805 .74412 .89777 1 


"39 


5 


56 


190 679 712 .1946 


518 547 725 • 


'53i 


827 392 830 


1132 


4 


57 


212 661 761 .1939 


540 528 776 . 


[524 


848 373 883 


1 126 


3 


58 


234 642 811 .1932 


562 509 827 . 


'5'7 


870 353 935 


1119 


2 


59 


256 623 860 .1925 


584 490 878 . 


[510 


891 334 988 


1113 


1 


60 


.64279 .76604 .83910 1.1918 


.65606 .75471 .86929 1. 


[504 


.66913 .74314 .90040 1 


1 106 

Tan. 




M, 




Cos. Sin. Cot. Tan, 


Cos. Sin. Cot. 1 


'an. 


Cos. Sin. Cot. 



50 



49° 



48 



240 NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
42° 43° 44° 



M. 


Sin. Cos. Tan, 


Cot. 


Sin. Cos. Tan. Cot. 


Sin. Cos. Tan. 


Cot. 


I 





•66913 .743H -90040 


[.1106 


.68200 .73135 .93252 I.0724 


.69466 .71934 -96569 


i-°355 


60 


i 


935 2 95 °93 


.1100 


221 116 306 .0717 


487 914 625 


•0349 


59 


2 


956 276 146 


.1093 


242 096 360 .0711 


508 894 681 


•0343 


58 


3 


978 256 199 


.1087 


264 076 415 .0705 


529 873 738 


•0337 


57 


4 


999 237 251 


.1080 


285 056 469 .0699 


549 853 794 


.0331 


56 


5 


.67021 .74217 .90304 


[.1074 


•68306 .73036 .93524 1.0692 


•69570 .71833 .96850 


1.0325 


55 


6 


043 198 357 


.1067 


327 016 578 .0686 


591 813 907 


.0319 


54 


7 


064 178 410 


.1061 


349 .72996 633 .0680 


612 792 963 


•0313 


53 


8 


086 159 463 


.1054 


370 976 688 .0674 


633 772 .97020 


•0307 


52 


9 


107 139 516 


.1048 


391 957 742 .0668 


654 752 076 


.0301 


5i 


10 


.67129 .74120 .90569 


[.1041 


.68412 .72937 .93797 1 .066 1 


•69675 •7 I 732 .97133 


1.0295 


50 


ii 


151 100 621 


.1035 


434 917 852 .0655 


696 711 189 


.0289 


49 


12 


172 080 674 


.1028 


455 897 906 .0649 


717 691 246 


.0283 


48 


13 


194 061 727 


.1022 


476 877 961 .0643 


737 671 302 


.0277 


47 


14 


215 041 781 


.1016 


497 857 .94016 .0637 


758 650 359 


.0271 


46 


15 


.67237 .74022 .90834 


[.1009 


.68518 .72837 .94071 1.0630 


.69779 .71630 .97416 


1.0265 


45 


16 


258 002 887 


.1003 


539 817 125 .0624 


800 610 472 


•0259 


44 


17 


280 .73983 940 


.0996 


561 797 180 .0618 


821 590 529 


.0253 


43 


18 


301 963 993 


.0990 


582 777 235 .0612 


842 569 586 


•0247 


42 


19 


323 944 .91046 


.0983 


603 757 290 .0606 


862 549 643 


.0241 


4i 


20 


.67344 73924 -91099 


[.0977 


.68624 .72737 .94345 I -°599 


.69883 .71529 .97700 


1-0235 


40 


21 


366 904 153 


.0971 


645 7 J 7 400 .0593 


904 508 756 


.0230 


39 


22 


387 885 206 


.0964 


666 697 455 .0587 


925 488 813 


.0224 


38 


23 


409 86$ 259 


.0958 


688 677 510 .0581 


946 468 870 


.0218 


37 


24 


430 846 313 


.0951 


709 657 565 .0575 


966 447 927 


.0212 


36 


25 


.6745 2 .73826 .91366 


[.0945 


.68730 .72637 .94620 1.0569 


.69987 .71427 .97984 


1.0206 


35 


26 


473 806 419 


.0939 


751 617 676 .0562 


.70008 407 .98041 


.0200 


34 


27 


495 7 8 7 473 


.0932 


77 2 597 73i -0556 


029 386 098 


.0194 


33 


28 


516 767 526 


.0926 


793 577 786 .0550 


049 366 155 


.0188 


32 


29 


538 747 580 


.0919 


814 557 841 .0544 


070 345 213 


.0182 


3i 


30 


•67559 -73728 .91633 


1-0913 


.68835 72537 .94896 1.0538 


.70091 .71325 .98270 


1 .0176 


30 


3i 


580 708 687 


.0907 


857 5 J 7 952 .0532 


112 305 327 


.0170 


29 


32 


602 688 740 


.0900 


878 497 .95007 .0526 


132 284 384 


.0164 


28 


33 


623 669 794 


.0894 


899 477 ' 062 .0519 


153 264 441 


.0158 


27 


34 


645 649 847 


.0888 


920 457 118 .0513 


174 243 499 


.0152 


26 


35 


.67666 .73629 .91901 


1 .088 1 


.68941 .72437 .95173 1.0507 


.70195 .71223 .98556 


1.0147 


25 


36 


688 610 955 


.0875 


962 417 229 .0501 


215 203 613 


.0141 


24 


37 


709 590 .92008 


.0869 


983 397 284 .0495 


236 182 671 


•0135 


23 


38 


730 57° 062 


.0862 


•69004 377 340 .0489 


257 162 728 


.0129 


22 


39 


75 2 55 1 ll6 


.0856 


° 2 5 357 395 .0483 


277 141 786 


.0123 


21 


40 


•67773-7353I -92170 


1.0850 


.69046 .72337 .95451 1.0477 


.70298 .71121 .98843 


1 .01 1 7 


20 


4i 


795 5 11 22 4 


.0843 


067 317 506 .0470 


319 100 901 


.0111 


19 


42 


816 491 277 


.0837 


088 297 562 .0464 


339 080 958 


.0105 


18 


43 


837 472 33i 


.0831 


109 277 618 .0458 


360 059 .99016 


.0099 


17 


44 


859 452 385 


.0824 


130 257 673 .0452 


381 039 073 


.0094 


16 


45 


.67880 .73432 ,92439 


1.0818 


.69151 .72236 .95729 1.0446 


.70401 .71019 .99131 


1.0088 


15 


46 


901 413 493 


.0812 


172 216 785 .0440 


422 .70998 189 


.0082 


14 


47 


923 393 547 


.0805 


193 196 841 .0434 


443 978 247 


.0076 


13 


48 


944 373 601 


.0799 


214 176 897 .0428 


463 957 304 


.0070 


12 


49 


965 353 655 


•0793 


235 J 5 6 952 -0422 


484 937 362 


.0064 


ri 


50 


•67987 -73333 -92709 


[.0786 


.69256 .72136 .96008 1.0416 


.70505 .70916 .99420 


[.0058 


10 


5i 


.68008 314 763 


.0780 


277 116 064 .0410 


525 896 478 


.0052 


9 


52 


029 294 817 


.0774 


298 095 120 .0404 


546 875 536 


.0047 


8 


53 


051 274 872 


.0768 


319 075 176 .0398 


567 855 594 


.0041 


7 


54 


072 254 926 


.0761 


340 055 232 .0392 


587 834 652 


.0035 


6 


55 


•68093 .73234 .92980 


»-°755 


.69361 .72035 .96288 1.0385 


.70608 .70813 .99710 ] 


.0029 


5 


56 


115 215.93034 


•0749 


382 015 344 .0379 


628 793 768 


.0023 


4 


57 


136 195 088 


.0742 


403 .7*995 400 .0373 


649 772 826 


.0017 


3 


58 


157 175 H3 


•0736 


424 974 457 -0367 


670 752 884 


.0012 


2 


59 


179 155 197 


.0730 


445 954 5 J 3 -°36i 


690 73i 942 


.0006 


1 


60 


.68200 .73135 .93252 


1.0724 


.69466 .71934 .96569 1.0355 


.70711 .70711 1.0000 i 


.0000 







Cos. Sin. Cot. 


Tan. 


Cos. Sin. Cot. Tan. 


Cos. Sin. Cot. 


Tan. 


mT 



47 ( 



46 c 



45 c 



AUXILIARY TABLE FOR SMALL ANGLES. 
0° 1° 2° 3° 4° 



241 



M. 


Sin. 


Tan. 


Sin. 


Tan. 


Sin. 


Tan. 


Sin. 


Tan. 


Sin. 


Tan. 


M. 




4. 


68 


4- 


58 


4- 


38 


4-< 


38 


4-< 


38 




o 


5575 


5575 


5553 


5 6l 9 


5487 


575 1 


5376 


5972 


5222 


6281 





i 


5575 


5575 


5552 


5620 


5485 


5754 


5374 


5976 


5219 


6287 


1 


2 


5575 


5575 


5551 


5622 


54 k 4 


5757 


5372 


598i 


5216 


6293 


2 


3 


5575 


5575 


5551 


5 6 23 


5482 


5760 


5370 


5985 


5213 


6299 


3 


4 


5575 


5575 


5550 


5625 


548i 


57 6 3 


5367 


5990 


5210 


6305 


4 


5 


5575 


5575 


5549 


5627 


5479 


5766 


5365 


5994 


5207 


6311 


5 


6 


5575 


5575 


5548 


5628 


5478 


5769 


5363 


5999 


5204 


6317 


6 


7 


5575 


5575 


5547 


5 6 3° 


5476 


5773 


536i 


6004 


5201 


6323 


7 


8 


5574 


5576 


5547 


5632 


5475 


5776 


5358 


6008 


5198 


6329 


8 


9 


5574 


5576 


5546 


5 6 33 


5473 


5779 


5356 


6013 


5195 


6335 


9 


IO 


5574 


5576 


5545 


5635 


547i 


5782 


5354 


6017 


5 J 92 


6341 


10 


ii 


5574 


5576 


5544 


5637 


547o 


5785 


535i 


6022 


5189 


6348 


11 


12 


5574 


5577 


5543 


5638 


5468 


5788 


5349 


6027 


5186 


6354 


12 


13 


5574 


5577 


5542 


5640 


5467 


5792 


5347 


6031 


5183 


6360 


13 


14 


5574 


5577 


5541 


5642 


5465 


5795 


5344 


6036 


5180 


6366 


14 


15 


5573 


5578 


5540 


5 6 44 


5463 


5798 


5342 


6041 


5 r 77 


6372 


15 


16 


5573 


5578 


5539 


5646 


5462 


5802 


5340 


6046 


5*73 


6379 


1 6 


17 


5573 


5578 


5539 


5648 


5460 


5805 


5337 


6051 


5*7° 


6385 


17 


18 


5573 


5579 


5538 


5 6 49 


5458 


5808 


5335 


6055 


5 l6 7 


6391 


18 


19 


5573 


5579 


5537 


5 6 5* 


5457 


5812 


5332 


6060 


5 l6 4 


6398 


19 


20 


5572 


558o 


5530 


5653 


5455 


5815 


533o 


6065 


5161 


6404 


20 


21 


5572 


558o 


5535 


5655 


5453 


5818 


5327 


6070 


5158 


6410 


21 


22 


5572 


558i 


5534 


5 6 57 


545i 


5822 


5325 


6075 


5 J 54 


6417 


22 


. 23 


5572 


5581 


5533 


5 6 59 


545° 


5825 


5322 


6080 


5*5* 


6423 


23 


24 


557i 


5582 


5532 


5661 


5448 


5829 


5320 


6085 


5148 


6430 


24 


25 


557i 


5583 


5531 


5663 


5446 


5833 


5317 


6090 


5*45 


6436 


25 


26 


557i 


5583 


553o 


5665 


5444 


5836 


53i5 


6095 


5 J 4i 


6443 


26 


27 


557o 


5584 


5529 


5668 


5443 


5840 


5312 


6100 


5138 


6449 


27 


28 


5570 


5584 


5527 


5670 


5441 


5843 


53io 


6105 


5*35 


6456 


28 


29 


557o 


5585 


5526 


5672 


5439 


5847 


5307 


6110 


5^2 


6462 


29 


30 


5569 


5586 


5525 


5674 


5437 


5851 


5305 


6116 


5128 


6469 


30 


31 


5569 


5587 


55 2 4 


5676 


5435 


5854 


5302 


6121 


5 I2 5 


6476 


3i 


32 


5569 


5587 


5523 


5 6 79 


5433 


5858 


5300 


6126 


5122 


6482 


32 


33 


5568 


5588 


5522 


5681 


543i 


5862 


5 2 97 


6131 


5118 


6489 


33 


34 


5568 


5589 


5521 


5683. 


5430 


5866 


5294 


6136 


5"5 


6496 


34 


35 


5567 


559o 


5520 


5685 


5428 


5869 


5292 


6142 


5112 


6503 


35 


36 


55 6 7 


559i 


5518 


5688 


5426 


5873 


5289 


6147 


5108 


6509 


36 


37 


5566 


5592 


55 J 7 


5690 


5424 


5877 


5286 


6152 


5 io 5 


6516 


37 


38 


5566 


5593 


55 l6 


5 6 93 


5422 


5881 


5284 


6158 


5101 


6523 


38 


39 


5566 


5593 


5515 


5 6 95 


5420 


5885 


5281 


6163 


5098 


6530 


39 


40 


5565 


5594 


5514 


5 6 97 


5418 


5889 


5278 


6168 


5°95 


6537 


40 


4i 


5565 


5595 


5512 


5700 


54i6 


5893 


5276 


6174 


5°9i 


6544 


4i 


42 


5564 


5596 


55" 


5702 


5414 


5897 


5273 


6179 


5088 


6551 


42 


43 


5564 


5598 


55io 


5705 


5412 


5900 


5270 


6185 


5084 


6557 


43 


44 


55 6 3 


5599 


55°9 


5707 


54io 


5905 


5268 


6190 


5081 


6564 


44 


45 


55 6 2 


5600 


5507 


57io 


5408 


5909 


5265 


6196 


5°77 


6571 


45 


46 


5562 


5601 


55°6 


5713 


5406 


5913 


5262 


6201 


5°74 


6578 


46 


47 


55 61 


5602 


5505 


57i5 


5404 


5917 


5 2 59 


6207 


5070 


6585 


47 


48 


55 61 


5 6 °3 


55°3 


5718 


5402 


592i 


5256 


6212 


5067 


6593 


48 


49 


556o 


5604 


55° 2 


5720 


5400 


5925 


5 2 54 


6218 


5063 


6600 


49 


50 


55 6 ° 


5 6o 5 


55 QI 


5723 


5398 


5929 


525 1 


6224 


5060 


6607 


50 


5i 


5559 


5607 


5499 


5726 


5396 


5933 


5248 


6229 


5056 


6614 


5i 


52 


5558 


5608 


5498 


5729 


5394 


5937 


5245 


6235 


5053 


6621 


52 


53 


5558 


5609 


5497 


573i 


5392 


5942 


5242 


6241 


5°49 


6628 


53 


54 


5557 


561 1 


5495 


5734 


5389 


5946 


5239 


6246 


5°45 


6635 


54 


55 


555 6 


5612 


5494 


5737 


5387 


5950 


5237 


6252 


5042 


6643 


55 


56 


555 6 


5 6l 3 


5492 


5740 


5385 


5955 


5234 


6258 


5038 


6650 


56 


57 


5555 


5 6l 5 


549i 


5743 


5383 


5959 


5231 


6264 


5°34 


6657 


57 


58 


5554 


5616 


5490 


5745 


538i 


5963 


5228 


6269 


5°3i 


6665 


58 


59 


5554 


5618 


5488 


5748 


5379 


5968 


5225 


6275 


5027 


6672 


59 

M. 


M. 


Sin. 


Tan. 


Sin. 


Tan. 


Sin. 


Tan. 


Sin. 


Tan. 


Sin. 


Tan. 



APPENDIX 



EXPLANATION OP TABLES* 
LOGARITHMS AND NATURAL FUNCTIONS 

1. Definition. — A logarithm of a number is the exponent of the power 
fco which a given number, called the base, must be raised to produce that 
number; or, in algebraic language, if a x = N, then log a JV=#, which is 
read "logarithm of N to the base a is equal x." 

2. Logarithms facilitate numerical calculations. For multiplying, 
dividing, raising to a power, or extracting a root, we have the following 
formulae, for a proof of which the student is referred to works on algebra : 

I. log A x B = log A + log B. 

II. log — = log A — log B. 
B 

III. log^. n = nlogA. 
IY. logVA =-logA. 

3. Logarithms were invented by John Napier, f and first published 
in 1614. Henry Briggs of England rendered the invention much more 
useful by employing 10 as the base. Our tables are computed to this 
base. We cannot here consider the method of computing! logarithms, 
for we are concerned mainly with the use of the tables. When there can 
be no uncertainty as to what base is used, it is not necessary to express 
the base ; thus, as 10 is the recognized base, instead of writing log 10 100 = 
2 we write simply log 100 = 2. 

*As Tables XVII, XVIII, XIX, and XX are copied, as elsewhere stated, from 
Professor Webster Wells's " Six-Place Logarithmic Tables," it is due him to say 
that this appendix was written by the author, Professor Wells being in no way 
responsible for this explanation of the tables. 

t Baron of Merchiston, Scotland. 

1 The formulae for this purpose may be found in works on trigonometry and higfier 

algebra. 

243 



244 PLANE SURVEYING 

4. The logarithm of any exact power of the base 10 is a whole number. 
This follows at once from the definition (Art. 1). The logarithm of any 
other power of the base is not an exact number, but consists of two parts, 
a whole number called the characteristic and a decimal part called the 
mantissa. 

For 10° = 1; hence log 1 = 

10 1 = 10, hence log - 10 = 1 

10 2 = 100, hence log 100 = 2 

10 3 = 1000, hence log 1000 = 3 ; 
and so on. 

Also 10 _1 = .1, hence log .1 = — 1 

10~ 2 = .01, hence log .01 = - 2 

10- 3 = .001, hence log .001 = - 3 ; 
and so on. 

From the above we see that 

the logarithm of a number between 1 and 10 lies between and 1, 

the logarithm of a number between 10 and 100 lies between 1 and 2, 

the logarithm of a number between 100 and 1000 lies between 2 and 3, 

and so on ; 

and 

the logarithm of a number between 1 and .1 lies between and — 1, 

the logarithm of a number between .1 and .01 lies between — 1 and — 2, 

the logarithm of a number between .01 and .001 lies between — 2 and — 3, 

and so on. 

5. The logarithm of a number less than 1 is negative, but it is so 
written that the mantissa (or decimal part) is always positive. For the 
number may be regarded as the product of two factors, one of which lies 
between 1 and 10, and the other is a negative power of 10 ; for example, 

.56 = 5.6 x 10" 1 , 
and log .56 = log 5.6 4- log 10 _1 

= - 1 + log 5.6 

= — 1 + .748188, which is written 
log .56 = 1.748188. 

The minus sign is placed above the characteristic to indicate that the sign 
belongs to it alone, while the mantissa is positive. 

Again, .024 = 10" 2 x 2.4. 

.-. log .024 = log 10- 2 + log 2.4 

= _ 2 + .380211 = 2.380211. 



APPENDIX 245 

6. From Arts. 4 and 5 r we derive the following rules for determining 
the characteristic of a logarithm : 

(a) If the number is greater than 1, the characteristic is positive and 
numerically one less than the number of figures in the integral part of the 
number. 

(b) If the number is less than 1, the characteristic is negative and numeri- 
cally one more than the number of zeros before the first significant figure of 
the decimal. 

Thus, the characteristic of log 427.32 = 2, 

the characteristic of log 9.7246 = 0, 

the characteristic of log .247 = — 1, 

the characteristic of log .00645 = — 3. 

Note. — The position of the decimal point affects the characteristic only, the man- 
tissa being the same for the same sequence of figures. For example, the mantissas of 
4582076, 45820.76, 458.2076, 4.582076, .0004582076, are alike. See Art. 12. 

7. The co-logarithm * of a number is the logarithm of the reciprocal of 
that number, and is obtained by subtracting the logarithm from O. f To 
avoid a negative characteristic, it is customary to subtract the logarithm, 
from 10 and take 10 from the result. 

Thus, log 2 = 0.301030; 

then colog 2 = (10 - 0.301030) - 10, 

= 9.698970 - 10. 
Again, log .002 = 3.301030. 

.-.' colog .002 = 12.698970 - 10, 
= 2.698970. 

The characteristic of a logarithm is always obtained by rule (Art. 6), 
and is seldom expressed in a table. The mantissa is obtained from the 
tables, as explained below. 

TABLE XVII 

8. This table, pages 163 to 178, contains the logarithms of numbers 
from 1 to 10,000, to six decimal places. The first three figures of the num- 
ber are given in the column headed "N," the fourth figure being found at 
the top or bottom of the columns to the right of the letter N. In the last 
column headed "D" is given the average difference between the successive 
mantissas in the row in which the difference is found. To make the page 
more open and thus lessen the strain on the eyes, the first two figures of 
the six are left out when their omission can cause no mistake. In looking 

* Sometimes called the arithmetical complement of the logarithm. 

t For, colog A = log— = log 1 — log A = — log A. 
A 



246 PLANE SURVEYING 

for a logarithm, if only the last four figures of the mantissa are found, 
the first two may be obtained from the nearest mantissa above, in the 
same column, which contains six figures. 

9. To find the Logarithm of a Number Less than 100. — We look in 
the "N" column for 100 times or 10 times the number according as the 
number is less than 10 or equal to or greater than 10, and take out the 
mantissa found in the " " column opposite. 

Thus, if the log 2 is required, we find on page 165, in the " " column 
opposite 200, the mantissa 301030 ; the characteristic determined by rule 
is 0. Hence, log 2 = .301030. 

So log 42 = 1.623249, the mantissa being given on page 169 in "0" 
column, opposite 420. 

10. To find the Logarithm of a Number of Three Figures. — Here the 

mantissa is found in the " " column opposite the given number. 
Thus, page 169, log 457 = 2.659916. 

11. To find the Logarithm of a Number of Four Figures. — Here the 

first three figures are found in the " N " column, the fourth is given at the 
top, or bottom, of the page, and the mantissa is taken out of the column 
containing the fourth figure and in the row in which the first three figures 
of the number are found. Thus, 

Page 172, log 5886 = 3.769820 ; 

Page 173, log 6400 = 3.806180 ; 

Page 174, log 78.26 = 1.893540. 

12. To find the Logarithm of a Number of Five or More Figures. — 

We must here use a process called interpolation* 

Kequired the logarithm of 43826. 

By our rule, we know at once that the characteristic is 4. Now the 
mantissa of 43826 is the same as the mantissa of 4382.6, and the difference 
between the mantissa of 4382 and 4382.6 is six-tenths the difference be- 
tween the mantissas of 4382 and 4383. 

From the table, page 169, we have 
mantissa of 4382 = .641672 

mantissa of 4383 = .641771 

difference = .000099 

and .6 of .000099 = .000059, 

the 4 which would appear in the seventh place being discarded, f 

* Interpolation is based on the assumption that between two successive mantissas 
the change in the mantissa is proportional to the change in the number. While this 
assumption is not strictly true, it gives a result sufficiently accurate in practice unless 
the logarithmic function of an angle very near to 0° or 90° is required. 

t When the fraction of a unit in such cases is less than .5, it is neglected ; if it is 
.5, or more than .5, it is to be taken as one unit, and the figure in the sixth place 
increased by 1, 



APPENDIX 247 

Therefore, adding this to the mantissa of 4382, we have 

.641672 + .000059 = .641731, 

and log 43826 = 4.641731. 

Note. —The difference between the two consecutive mantissas is usually obtained 
mentally, or else simply taken from the "D" column in the row in which the first 
three figures of the number are found. It is not necessary, or usual, to write this 
difference as a decimal, as it is always to be applied to the last figures of the mantissa. 
In this case, we should say .6 x 99 = 59, and this, added (mentally) to .641672, gives 
.641731, the mantissa sought. 

Required log 675746. 

Here, remembering that the mantissa of 6757.46 is the same as the 
mantissa of 675746, we have, from page 173, 

mantissa of 6757 = .829754, 

mantissa of 6758 = .829818. 

Difference (taken from " D " column) = 64. 

Here we take .46 of 64 = 29.44, which we call 29. Adding 29 to 
.329754, we have ]og 675746 = 5<829783> 

Required log .00017428. 

The mantissa is the same as the mantissa of 1742.8, and is therefore 
equal to the mantissa of 1742 plus .8 of the difference between the mantissas 
of 1742 and 1743 (that is, page 165, .8 x 249 = 199). 

.-. log .00017428 = 4.241247. 

Note. — If the given number contained seven significant figures, then we should 
have to take so many thousandths of the tabular difference. Thus, page 12, 

log 724.6873 = 2.860098 + .873 x 60 = 2.860150. 

To find the Antilogarithm, that is, the number corresponding to a given 
logarithm. 

13. If the exact mantissa can be found in the table, then the correspond- 
ing number is taken out at once, the first three figures being found in the 
" NT " column in the row with the given mantissa, and the fourth figure at 
the top (bottom) of the column in which the mantissa occurs. 

The characteristic determines the position of the decimal point in the 
antilogarithm, according to the following rules : 

If the characteristic is positive, point off one more place than the number 
denoted by the characteristic. 

If the characteristic is negative, the antilogarithm is entirely decimal and 
the number of zeros immediately after the decimal point is one less than the 
number denoted by the characteristic. 



248 PLANE SURVEYING 

Examples : 

If log x = 2.147676, find x. On page 164, we find this exact mantissa 
in the row with 140 (in " N " column) and in the column headed 5. Hence, 
the antilogarithm is 140.5, the characteristic, 2, determining the position 
of the decimal point. 

Again, if logx = 4.329398, then, page 165, x = .0002135, for the man- 
tissa is found in the row with 213 and in column " 5," and the character- 
istic shows that there are three zeros before the first significant figure of 
the decimal. 

14. If the mantissa of the given logarithm cannot be found in the 
table, then we must interpolate. 

Given log x = 2.306530, to find x. 

Here the mantissa lies between .306425 and .306639, which are two 
consecutive mantissas given in the table, page 165. Hence, the antiloga- 
rithm lies between 202.5 and 202.6. The difference between the two con- 
secutive mantissas is .306639 - .306425 = .000214, or, as we say, 214. 
[Here the " D " column gives 215, which would give practically the same 
result.] The difference between the given mantissa, .306530, and the 
smaller of these, .306425, is 105. Now, we assume* that if a difference of 
214 in the mantissa makes a difference of 1 in the fourth place of the anti- 
logarithm, then a difference of 105 in the mantissa will make a difference 
of i^-| of 1 in the antilogarithm. 

Hence, Jff of 1 = 0.4907 is to be annexed to 2025, making 20254907. 

.-. x = 202.54907. 

Note. — In practice, the decimal here would probably not have been carried beyond 
the third place. 

Again, given log x = 3.837241, find x. 

Here, page 173, the mantissa lies between .837210 and .837273, and 
hence (ignoring the characteristic for the moment) the antilogarithm lies 
between 6874 and 6875. The tabular difference is 63, and the difference 
between the given mantissa .837241 and .837210 is 31. 

Hence, we annex to 6874 

fiofl = .49. 

.-. a = .00687449. 

As a further illustration of the application of logarithms and the use 
of the tables, we give in the next article a few examples. 

15. Examples. 

(1) Find the value of x, if x = 82473 x .0723. 

Here (Art. 2) log sc=log 82473 4- log .0723, and the work is conveniently 

arranged as follows : 

* Footnote, page 246. 



APPENDIX 249 

Page 176, log 82473 = 4.916312 

Page 174, log .0723 = 2.859138 

Art. 2, .-. log x = 3.775450 

Page 172, .-. x = 5962.794. 

Note.— Multiplying 82,473 by .0723 by the ordinary process, we get x = 5962.7979, 
which shows that, by employing logarithms, we get a result differing from the true 
product by .0039. The result, then, as might be inferred, is an approximate one ; but 
logarithms are only used where such slight discrepancies make no practical error. 

(2) Find x, if x = 472068 -*- 34.2. 
Here log x = log 472068 - log 34.2. 

Page 8, log 472068 = 5.674005 

Page 6, log 34.2 = 1.534026 

Subtracting, log x = 4.139979 

.-. x = 13803.17. 
Note. — Instead of subtracting the logarithm, we may add the cologarithm of 34.2 } 

thUS(Art ' 7): log 472068 = 5.674005 ' 

colog 34.2 = 8.465974 - 10 
Adding, log x = 4. 139979 

(3) Find x, if « = 384 x .0024 x 782.94. 

v ' .07824 x 47 

Here log 384 = 2.584331 

log .0024= 3.380211 
log 782.94= 2.893728 
colog .07824 = 11.106571 - 10 
colog 47 = 8.327902 - 10 
.-., adding, log x = 22.292743 - 20 

or logo;= 2.292743. 

.-. x = 196.219. 

Note. — A negative number has no common logarithm. If such numbers occur in 
practice, they are treated as positive numbers, and the sign of the result is determined 
irrespective of the logarithmic work. For example, if the value of — 24782 x + 78.702 
is to be determined by logarithms, we treat both numbers as positive in getting their 
logarithms, and we give the result the minus sign, because the product of a + quantity 
by a — quantity is negative. 

TABLE XVIII 

16. Table XVIII gives the logarithmic sines, cosines, tangents, and 
cotangents from 0° to 90°. The formulae 

sin (90° - A) = cos A, cos (90° — A) = sin A, 

tan (90° - A) = cot A, cot (90° - A) = tan A, 



250 PLANE SURVEYING 

are used to reduce the size of such a table one-half, as will now appear. 
At the top of the pages the degree numbers run from 0° to 44°, while at 
the bottom, they run in reverse order from 45° to 90° ; and at the bottom 
the cosine, sine, cotangent, and tangent columns correspond respectively 
to the sine, cosine, tangent, and cotangent at the top. If the angle is less 
than 45°, the function is read at the top of the page ; if it is greater than 
45°, the function is found at the bottom of the page. The minutes for the 
functions at the top are found in the first column and run down, while the 
minutes for the functions at the bottom are given in the last column and 
run up. If seconds occur, interpolation is used. 

Where the characteristic would be negative, 10 is added to make it 
positive, and this positive characteristic is given in the table. 

17. To find the logarithmic function ivhen the angle contains degrees and 
minutes and no seconds. 

Here the logarithmic function is taken out at once from the table ; for 
example : 

Page 188, log sin 8° 20' = 9.161164. 

Page 181, , log cos 88° 10' = 8.505045. 

Page 197, log tan 17° 53' = 9.508759. 

Page 221, log cot 41° 14' = 0.057267. 

Caution. — When the angle is greater than 45°, read the function at 
the bottom of the page, and read the minutes on the right-hand side of the 
page. 

18. To find the logarithmic function when the angle contains seconds. 

Here we must interpolate (see footnote, page 246). To assist in this 
interpolation, the difference in the last figures of the logarithm for 1" is 
given.* As the tangent and cotangent vary alike, one column of differ- 
ences is sufficient for both, and this, in our table, is put between the 
tangent and cotangent columns. 

(1) To find log sin 20° 34' 18". 

Here, page 200, log sin 20° 34' = 9.545674, as found in the sine column 
opposite 34 minutes. 

Difference for 1" = 5.62. 

.-. difference for 18" = 18 x 5.62 = 101, 

and 9.545674 + 101 = 9.545775, 

or log sin 20° 34' 18" = 9.545775. 

* It will "be noticed that these differences do not stand in the same horizontal row 
with the logarithms, but opposite the intervals between consecutive logarithms. With 
the degrees at the top of the page, the difference next below should be taken ; with the 
degrees at the bottom of the page, the difference next above. 



APPENDIX 251 

(2) To find log cos 41° 10'' 47". 

Page 221, log cos 41° 10' = 9.876678 

Difference for 47" = 47 x 1.83 = 86 

.-., subtracting, log cos 41° 10' 47" = 9.876592 

Here the difference for 47" is subtracted, for the cosine decreases as 
the angle increases. 

Note. — Observe that the sine and tangent increase, while the cosine and cotangent 
decrease, as the angle increases. 

(3) To find log tan 54° 38' 52". 

As the angle is greater than 45°, we read from the botton, page 215. 

log tan 54° 38' = .148871 

Difference = 52 x 4.47 = 232 

.-., adding, log tan 54° 38' 52" = .149103 

(4) To find the log cot 11° 52' 44". 

Page 191, log cot 11° 52' = .677521 

Difference = 44 x 10.45 = 460 



.-., subtracting, log cot 11° 52' 44" = .677061 

To find the angle token the logarithmic function is given. 

19. If the exact logarithmic function is found in the table, then the 
angle may be at once written down by taking the degrees from the top or 
bottom, as the case may be, and the minutes from the minute column on 
the left or right of the page. 

Thus, if log sin A = 9.303979, 
then, page 191, A = 11° 37'. 

If log cot B = 9.318697, 
then, page 191, B = 78° 14'. 

20. If the exact logarithmic function cannot be found in the table, 
then the angle is found by interpolation, as in the following examples : 

(1) Given log sin A = 9.476721, to find A. 

Here (page 197) the mantissa is contained between the mantissas cor- 
responding to 17° 26' and 17° 27' ; hence, evidently, A = 17° 26' + some 
seconds. Now, 

Mantissa for log sin 17° 26' = .476536 

Mantissa for log sin A = .476721 

Difference = 185 

In the difference column we find that the logarithmic sines at this point 
increase at the rate of 6.70 per second. Hence, a difference of 185 would 

correspond to — = 28", to the nearest second. 
6 ' 7 .-. A = 17° 26' 28". 



252 PLANE SURVEYING 

(2) Given log cos A = 9.720238, to find A. 

We see (page 211) that the given logarithm lies between the logarithmic 

cosine of 58° 19' and 58° 20'. 

log cos 58° 19' = 9.720345 

log cos A = 9.720238 

Difference = 107 

D for 1" = 3.42. 

107 
.-. the number of seconds to be added = — — =31, and therefore ^4 = 58° 

19' 31". 3 - 2 

(3) Given log tan A = 9.990201, to find A. 

Here (page 224) the given logarithm lies between 9.990145 and 9.990398, 
and therefore the angle lies between 44° 21' and 44° 22'. 

log tan 44° 21' = 9.990145 
log tan .4=9.990201 

Difference = 56 

56 
Seconds to be added = = 13. 

4.22 

.-. A = 44° 21' 13". 

Note. — In practice, the difference, such as 56 in Ex. 3, is usually obtained mentally, 
and thus the work is greatly expedited. 

(4) Given log cot A = 0.327941, to find A. 

Here the angle evidently (page 205) lies between 25° 10' and 25° 11'. 
Difference between log cot 25° 10' and log cot A is 96, and Z> = 5.47. 

96 -r- 5.47 = 18", to be added. 

.-. .4 = 25° 10' 18". 

21. Secant and Cosecant. — If the logarithmic secant and logarithmic 
cosecant are required, we express these functions in terms of the cosine 
and sine by the formulae : 

sec x = , cosec x = — — , 

cos x sin x 

whence log sec x = colog cos x. 

log cosec x = colog sin x. 

22. Angles Greater than 90°. 

Angles in the second quadrant are reduced to acute angles by means 
of the formulae : 

sin (180° — x) = sin x ; tan (180° — a;) = — tan x ; 

cos (180° — x) = — cos x ; cot (180° — x) = — cot x. 

Angles greater than 180° are also readily reduced to acute angles by 
the proper trigonometric formulae. 



APPENDIX 253 

Note. — Negative functions. Strictly speaking, a negative angle or a negative 
function has no logarithm. When they occur, we treat them as positive as far as the 
logarithmic work is concerned. Compare note under Ex. 3, Art. 15. Also see Ex. 1, 
Art. 23. To call attention to the fact that the function is negative, the letter "n" is 
sometimes written after the logarithmic function ; thus, 

since cos 120° 40' = - cos (180° - 120° 40') = - cos 59° 20', 

log cos 120° 40' = 9.707606 n - 10. 

23. Examples. 

(1) If x = 123.8 cos 120° 40', find x. 

log 123.8 = 2.092721 
log cos 120° 40' = 9.707606 n - 10 
.-. loga? = 1.800327 n 
.-. x = -63.143. 

/9\ Tfc-n a 40.36sin41°10'36" fl , A 

(2) K Sm ^ = 12132 ' find ^ 

log 40.36 = 1.605951 
log sin 41° 10' 36" = 9.818478 - 10 
colog 124.32 = 7.905459 - 10 
.-. log sin ^4 = 9.329888 -10 

.-. A = 12° 20' 30". 

,os jo ' 4721.6 x tan 0° 40' 24" ~ , 

(o) It x = , find x. 

W 348 x cot 20° 52' ' 

log 4721.6= 3.674089 

* log tan 0° 40' 24"= 8.070096-10 

colog 348= 7.458421-10 

colog cot 20° 52' = 9.581149 - 10 
log x = 28.783755 - 30 

= 2.783755. 

Whence x = 0.06078. 



TABLE XIX 

24. This table speaks for itself, and detailed explanations seem un- 
necessary. Here the values of the natural functions for each minute from 
0° to 90° are given to five decimal places. 

If it is desired to have the values for fractions of a minute, we interpo- 
late just as in the logarithmic tables, only in this table the tabular dif- 

* The logarithmic function of this small angle could be found with a greater degree 
of accuracy by the use of Table XX. See Art. 27. 



254 PLANE SURVEYING 

ference for 1" is not given, but must be computed. It is one-sixtieth of 
the difference between two consecutive values of the function. 
For example, let the natural sine of 20° 34' 37" be required. 

Page 232, sin 20° 34' = .35130 

sin 20° 35 '= .35157 
Difference for 60" = 27 
.-. Difference for 37" = f-J x 27 = 17. 

.-. sin 20° 34' 37" = .35130 + .00017 = .35147. 

25. Table XIX as a Traverse Table. — As noted in Art. 151, that part 
of this table in which the sines and cosines are given may be used as a 
traverse table. For the distance unity, the sine gives the " departure " of a 
course, the cosine, its " latitude." 

Examples. 

(1) Eequired the departure and latitude of a course whose bearing is 
N. 21° 40' E. and whose length is 4 chains. On page 201, we find 

sin 21° 40' = .36921 = departure for distance 1. 

cos 21° 40' = .92935 = latitude for distance 1. 

.'. Departure of given course = 4 x .36921 = 1.48. 

Latitude of given course = 4 x .92935 = 3.72. 

(2) Eequired the latitude and departure of a course AB, of length 
21.36 chains, whose bearing is S. 82° 20' W. 

Here, page 228, 

sin 82° 20' = .99106 = departure for distance 1. 
cos 82° 20' = .13341 = latitude for distance 1. 
.*. Departure of AB = 21.36 x .99106 = 21.17. 
Latitude of AB = 21.36 x .13341 = 2.85. 

TABLE XX 
Auxiliary Table for Small Angles 

26. This table gives under the heads " Sin " and " Tan," respectively, 
the values of the two expressions. 

10 + log sin x — log x and 10 + log tan x — log x, x being expressed in 
seconds for each minute from 0° to 4° 59'. 

The object of the table is to find the logarithmic sines and tangents of 
angles between 0° and 5° (and therefore of cosines and cotangents between 
90° and 85°) to a greater degree of accuracy than is possible from Table 
XVIII. Within the same limits, it may of course be used in obtaining 
the angle from a given logarithmic function. 



APPENDIX 255 

27. To find the logarithmic sine- or tangent of an angle between 0° and 5°. 
Find from the auxiliary table the logarithm corresponding to the given 

function, add to the result the logarithm of the number of seconds in the 
angle, and write — 10 after the mantissa. 

Example. — Find log sin 2° 35' 26". 

The logarithms (from Table XX) corresponding to sin 2° 35' and 
sin 2° 36' are 4.685428 and 4.685426, and their difference is 2. 

Hence, -§£ of 2 = 1, nearly, is to be subtracted from 4.685428, giving 
4.685427. 

Now 2° 35' 26" = 9326". 

4.685427 - 10 
log 9326 = 3 .969695 
.-. log sin 2° 35' 26" = 8.655122 - 10 
The result by Table XVIII is 8.655121 - 10. 

28. To find the angle corresponding to a given logarithmic sine or tangent 
ivhen between 0° and 5°. 

Find from Table XVIII the angle corresponding to the given logarithm, 
to the nearest second. 

Take from Table XX the logarithm corresponding to this angle. 

Subtract the resxdt from the given logarithm, and find the number corre- 
sponding to the difference, giving the required angle in seconds. 

Example. — Find the angle whose log tan = 8.021348. 
From Table XVIII, the angle corresponding is 0° 36' 7" to the nearest 
second. 

From Table XX, the logarithm corresponding to 0° 36' 7" is 

4.685591 - 10. 

8.021348 - 10 
4.685591 - 10 
3.335757 

Number corresponding to this = 2166.49. 
.-. required angle = 2166.49" = 0° 36' 6.49". 

Note. — Remembering that the tangent and cotangent are reciprocals, we see that 
this table may readily serve to determine the cotangent of an angle between 0° and 
5° and the tangent of an angle between 85° and 90°, or the angle corresponding in the 
same cases. 



COLLEGE ALGEBRA 

By WEBSTER WELLS, S.B., 

Professor of Mathematics in the Massachusetts Institute 
of Technology. 



The first eighteen chapters have been arranged with reference 
to the needs of those who wish to make a review of that portion 
of Algebra preceding Quadratics. While complete as regards 
the theoretical parts of the subject, only enough examples are 
given to furnish a rapid review in the classroom. 

Attention is invited to the following particulars on account of 
which the book may justly claim superior merit : — 

The proofs of the five fundamental laws of Algebra — the Com- 
mutative and Associative Laws for Addition and Multiplication, and 
the Distributive Law for Multiplication — for positive or negative 
integers, and positive or negative fractions ; the proofs of the 
fundamental laws of Algebra for irrational numbers ; the proof of 
the Binomial Theorem for positive integral exponents and for 
fractional and negative exponents ; the proof of Descartes's Rule 
of Signs for Positive Roots, for incomplete as well as complete 
equations ; the Graphical Representation of Functions ; the so- 
lution of Cubic and Biquadratic Equations. 

In Appendix I will be found graphical demonstrations of the 
fundamental laws of Algebra for pure imaginary and complex 
numbers ; and in Appendix II, Cauchy's proof that every equa- 
tion has a root. 



Half leather. Pages, vi + 5j8. Introduction price, $7.50. 
Part II, be ginning with Quadratics. 341 pages. Introduction price, $1.32. 

D. C. HEATH & CO., Publishers, Boston, New York, Chicago 



DEC 15 1913 -5C, . 



THEORY OF EQUATIONS 

By SAMUEL MARX BARTON, Ph.D., 
Professor of Mathematics in the University of the South. 



In this treatise the author aims to give the elements of Deter- 
minants and the Theory of Equations in a form suitable, both in 
amount and quality of matter, for use in undergraduate courses. 
The work is readily intelligible to the average student who has be- 
come proficient in algebra and the elements of trigonometry. 
The use of the calculus has been purposely avoided. While the 
presentation of the subject has necessarily been condensed to 
suit the requirements of college courses, great pains has been 
taken not to sacrifice clearness to brevity. It is a short treatise, 
but not a syllabus. 

Part I treats of Determinants. The chapters give the funda- 
mental theorems, with examples for illustration ; applications and 
special forms of determinants, followed by a collection of care- 
fully selected examples. 

Part II treats of the Theory of Equations proper, with chapters 
upon complex numbers, properties of polynomials, general 
properties of equations, relations between roots and coefficients, 
symmetric functions, transformation of equations, limits of the 
roots of an equation, separation of roots, elimination, solution of 
numerical equations. Almost every theorem is elucidated by 
$he complete solution of one or more representative examples. 



Clotb. Tages,x+ 198. Introduction price, $1.50. 



D. C. HEATH & CO., Publishers, Boston, New York, Chicago 

-31T** 



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